Displaying 1-10 of 72 results found.
Expansion of (7-14*x+6*x^2)/((1-x)*(2*x^2-4*x+1)); related to the binomial transform of Pell numbers A000129 (see formula and comment for A007070).
+20
0
7, 21, 69, 233, 793, 2705, 9233, 31521, 107617, 367425, 1254465, 4283009, 14623105, 49926401, 170459393, 581984769, 1987020289, 6784111617, 23162405889, 79081400321, 270000789505, 921840357377, 3147359850497, 10745758687233
COMMENTS
If g.f. (x^6+5*x^4+6*x^2+1)/(x^7+6*x^5+10*x^3+4*x) is expanded, where (x^6+5*x^4+6*x^2+1) and (x^7+6*x^5+10*x^3+4*x) are the 7th and 8th Fibonacci polynomials, respectively, the sequence: [0, 7/8, 0, -21/16, 0, 69/32, 0, -233/64, 0, 793/128, 0, -2705/256, ] is returned. (a(n)) is seen to be the numerators of the bisection of this sequences, apart from signs.
FORMULA
a(n+1) - a(n) = A007070(n+2), a(n) - 2*a(n+1) + a(n+2) = A007052(n+3) (Number of order consecutive partitions of n), a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n) = A003480(n+4), a(n+2) - a(n) = A111567(n+3)
MAPLE
with(combinat, fibonacci): seq(fibonacci(i, x), i=1..15); [[generates sequence of Fibonacci polynomials]]
Number of order-consecutive partitions of n.
(Formerly M2847)
+10
89
1, 3, 10, 34, 116, 396, 1352, 4616, 15760, 53808, 183712, 627232, 2141504, 7311552, 24963200, 85229696, 290992384, 993510144, 3392055808, 11581202944, 39540700160, 135000394752, 460920178688, 1573679925248
COMMENTS
After initial terms, first differs from A291292 at a(6) = 1352, A291292(8) = 1353. Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 3) is "size of raises in pot-limit poker, one blind, maximum raising".
Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 3, s(2n+1) = 4. - Herbert Kociemba, Jun 12 2004 Equals the INVERT transform of (1, 2, 5, 13, 34, 89, ...). - Gary W. Adamson, May 01 2009 a(n) is the number of compositions of n when there are 3 types of ones. - Milan Janjic, Aug 13 2010 a(n)/a(n-1) tends to (4 + sqrt(8))/2 = 3.414213.... Gary W. Adamson, Jul 30 2013 Number of words of length n over {0,1,2,3,4} in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017 Also the number of unimodal sequences of length n + 1 covering an initial interval of positive integers, where a sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. For example, the a(0) = 1 through a(2) = 10 sequences are:
(1) (1,1) (1,1,1)
(1,2) (1,1,2)
(2,1) (1,2,1)
(1,2,2)
(1,2,3)
(1,3,2)
(2,1,1)
(2,2,1)
(2,3,1)
(3,2,1)
Missing are: (2,1,2), (2,1,3), (3,1,2).
Conjecture: Also the number of ordered set partitions of {1..n + 1} where no element of any block is greater than any element of a non-adjacent consecutive block. For example, the a(0) = 1 through a(2) = 10 ordered set partitions are:
{{1}} {{1,2}} {{1,2,3}}
{{1},{2}} {{1},{2,3}}
{{2},{1}} {{1,2},{3}}
{{1,3},{2}}
{{2},{1,3}}
{{2,3},{1}}
{{3},{1,2}}
{{1},{2},{3}}
{{1},{3},{2}}
{{2},{1},{3}}
a(n-1) is the number of hexagonal directed-column convex polyominoes having area n (see Baril et al. at page 4). - Stefano Spezia, Oct 14 2023
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n+1) = 4a(n) - 2a(n-1).
G.f.: (1-x)/(1-4x+2x^2).
Binomial transform of Pell numbers 1, 2, 5, 12, ... ( A000129). a(n) = ( A035344(n)+1)/2; a(n) = (2+sqrt(2))^n(1/2+sqrt(2)/4)+(2-sqrt(2))^n(1/2-sqrt(2)/4). - Paul Barry, Jul 16 2003 Second binomial transform of (1, 1, 2, 2, 4, 4, ...). a(n) = Sum_{k=1..floor(n/2)}, C(n, 2k)*2^(n-k-1). - Paul Barry, Nov 22 2003 a(n) = ( (2-sqrt(2))^(n+1) + (2+sqrt(2))^(n+1) )/4. - Herbert Kociemba, Jun 12 2004 a(n) = both left and right terms in M^n * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 1 2 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A007070(n) a(n)]. E.g., a(3) = 34. M^3 * [1 1 1] = [34 48 34] (center term is A007070(3)). - Gary W. Adamson, Dec 18 2004 The i-th term of the sequence is the entry (2, 2) in the i-th power of the 2 X 2 matrix M = ((1, 1), (1, 3)). - Simone Severini, Oct 15 2005 E.g.f. : exp(2x)(cosh(sqrt(2x)+sinh(sqrt(2)x)/sqrt(2). - Paul Barry, Nov 20 2003 If p[i]=Fibonacci(2i-1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010 a(n-1) = Sum_{k=-floor(n/4)..floor(n/4)} (-1)^k*binomial(2*n,n+4*k)/2. - Mircea Merca, Jan 28 2012 G.f.: G(0)*(1-x)/(2*x) + 1 - 1/x, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - (1-x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013 a(n) = 3*a(n-1) + a(n-2) + a(n-3) + a(n-4) + ... + a(0). - Gary W. Adamson, Aug 12 2013 a(n) = a(-2-n) * 2^(n+1) for all n in Z. - Michael Somos, Jan 25 2017
EXAMPLE
G.f. = 1 + 3*x + 10*x^2 + 34*x^3 + 116*x^4 + 396*x^5 + 1352*x^6 + 4616*x^7 + ...
MATHEMATICA
a[n_]:=(MatrixPower[{{3, 1}, {1, 1}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *) a[ n_] := ((2 + Sqrt[2])^(n + 1) + (2 - Sqrt[2])^(n + 1)) / 4 // Simplify; (* Michael Somos, Jan 25 2017 *) unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Union@@Permutations/@allnorm[n], unimodQ]], {n, 6}] (* Gus Wiseman, Mar 06 2020 *)
PROG
(PARI) {a(n) = real((2 + quadgen(8))^(n+1)) / 2}; /* Michael Somos, Mar 06 2003 */ (Magma) [Floor((2+Sqrt(2))^n*(1/2+Sqrt(2)/4)+(2-Sqrt(2))^n*(1/2-Sqrt(2)/4)): n in [0..30] ] ; // Vincenzo Librandi, Aug 20 2011
CROSSREFS
Cf. A000129, A000670, A001523, A001653, A007068, A035344, A060223, A075271, A227038, A291292, A328509, A332577, A332743, A332873.
a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - 2*a(n-2), n >= 2.
(Formerly M1644)
+10
52
1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, 367424, 1254464, 4283008, 14623104, 49926400, 170459392, 581984768, 1987020288, 6784111616, 23162405888, 79081400320, 270000789504, 921840357376, 3147359850496
COMMENTS
a(n)/a(n-1) approaches 2 + sqrt(2). - Zak Seidov, Oct 12 2002 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 4, s(2n) = 4. - Herbert Kociemba, Jun 12 2004 a(k) = [M^k]_{2,2}, where M is the following 3 X 3 matrix: M = [1,1,1;1,2,1;1,1,1]. - Simone Severini, Jun 11 2006 a(n-1) counts permutations pi on [n] for which the pairs {i, pi(i)} with i < pi(i), considered as closed intervals [i+1,pi(i)], do not overlap; equivalently, for each i in [n] there is at most one j <= i with pi(j) > i. Counting these permutations by the position of n yields the recurrence relation. - David Callan, Sep 02 2003 Counts all paths of length (2*n), n >= 0, starting at the initial node on the path graph P_7, see the second Maple program. - Johannes W. Meijer, May 29 2010 Let U_1 and U_3 be the unit-primitive matrices (see [Jeffery])
U_1 = U_(8,1) = [(0,1,0,0); (1,0,1,0); (0,1,0,1); (0,0,2,0)] and
U_3 = U_(8,3) = [(0,0,0,1); (0,0,2,0); (0,2,0,1); (2,0,2,0)]. Then A006012(n) = (1/4) * Trace(U_1^(2*n)) = (1/2^(n+2)) * Trace(U_3^(2*n)). (See also A084130, A001333.) (End) Pisano period lengths: 1, 1, 8, 1, 24, 8, 6, 1, 24, 24, 120, 8, 168, 6, 24, 1, 8, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012 Conjecture: With offset 1, a(n) is the number of permutations on [n] with no subsequence abcd such that (i) bc are adjacent in position and (ii) max(a,c) < min(b,d). For example, the 4 permutations of [4] not counted by a(4) are 1324, 1423, 2314, 2413. - David Callan, Aug 27 2014 The conjecture of David Callan above is correct - with offset 1, a(n) is the number of permutations on [n] with no subsequence abcd such that (i) bc are adjacent in position and (ii) max(a,c) < min(b,d). - Yonah Biers-Ariel, Jun 27 2017 A production matrix for the sequence is M =
1, 1, 0, 0, 0, 0, ...
1, 0, 3, 0, 0, 0, ...
1, 0, 0, 3, 0, 0, ...
1, 0, 0, 0, 3, 0, ...
1, 0, 0, 0, 0, 3, ...
...
Take powers of M, extracting the upper left terms; getting the sequence starting: (1, 1, 2, 6, 20, 68, ...).
(End)
The sequence is the INVERT transform of the powers of 3 prefaced with a "1": (1, 1, 3, 9, 27, ...) and is N=3 in an infinite of analogous sequences starting:
N=1 ( A000079): 1, 2, 4, 8, 16, 32, ... N-2 ( A001519): 1, 2, 5, 13, 34, 89, ... N=3 ( A006012): 1, 2, 6, 20, 68, 232, ... N=4 ( A052961): 1, 2, 7, 29, 124, 533, ... N=5 ( A154626): 1, 2, 8, 40, 208, 1088, ... N=6: 1, 2, 9, 53, 326, 2017, ...
...
(End)
Number of permutations of length n > 0 avoiding the partially ordered pattern (POP) {1>2, 1>3, 4>2, 4>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first and fourth elements are larger than the second and third elements. - Sergey Kitaev, Dec 08 2020 a(n-1) is the number of permutations of [n] that can be obtained by placing n points on an X-shape (two crossing lines with slopes 1 and -1), labeling them 1,2,...,n by increasing y-coordinate, and then reading the labels by increasing x-coordinate. - Sergi Elizalde, Sep 27 2021
REFERENCES
D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms. Birkhäuser, Boston, 3rd edition, 1990, p. 86.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.4.8 Answer to Exer. 8.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f.: (1-2*x)/(1 - 4*x + 2*x^2).
Binomial transform of A001333. E.g.f. exp(2x)cosh(x*sqrt(2)). - Paul Barry, May 08 2003 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*2^(n-k) = Sum_{k=0..n} binomial(n, k)*2^(n-k/2)(1+(-1)^k)/2. - Paul Barry, Nov 22 2003 (typo corrected by Manfred Scheucher, Jan 17 2023) a(n) = ((2+sqrt(2))^n + (2-sqrt(2))^n)/2.
G.f.: G(0) where G(k)= 1 + 2*x/((1-2*x) - 2*x*(1-2*x)/(2*x + (1-2*x)*2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2012 G.f.: G(0)*(1-2*x)/2, where G(k) = 1 + 1/(1 - 2*x*(4*k+2-x)/( 2*x*(4*k+4-x) + 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 27 2014 a(n) = A265185(n) / 4, connecting this sequence to the simple Lie algebra B_4. - Tom Copeland, Dec 04 2015
MAPLE
with(GraphTheory): G:=PathGraph(7): A:= AdjacencyMatrix(G): nmax:=24; n2:=2*nmax: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1, k], k=1..7); od: seq(a(2*n), n=0..nmax); # Johannes W. Meijer, May 29 2010
MATHEMATICA
LinearRecurrence[{4, -2}, {1, 2}, 50] (* or *) With[{c=Sqrt[2]}, Simplify[ Table[((2+c)^n+(3+2c)(2-c)^n)/(2(2+c)), {n, 50}]]] (* Harvey P. Dale, Aug 29 2011 *)
PROG
(PARI) {a(n) = real(((2 + quadgen(8))^n))}; /* Michael Somos, Feb 12 2004 */ (PARI) {a(n) = if( n<0, 2^n, 1) * polsym(x^2 - 4*x + 2, abs(n))[abs(n)+1] / 2}; /* Michael Somos, Feb 12 2004 */ (PARI) Vec((1-2*x)/(1-4*x+2*x^2) + O(x^100)) \\ Altug Alkan, Dec 05 2015 (Magma) [n le 2 select n else 4*Self(n-1)- 2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 05 2011 (Haskell)
a006012 n = a006012_list !! n
a006012_list = 1 : 2 : zipWith (-) (tail $ map (* 4) a006012_list)
(map (* 2) a006012_list)
(Python)
l = [1, 2]
for n in range(2, 101): l.append(4 * l[n - 1] - 2 * l[n - 2])
CROSSREFS
Cf. A140070, A152252, A024175, A030436, A094803, A003480, A265185, A000079, A001519, A052961, A154626.
a(n) = 4*a(n-1) - 2*a(n-2) (n >= 3).
(Formerly M1763)
+10
43
1, 2, 7, 24, 82, 280, 956, 3264, 11144, 38048, 129904, 443520, 1514272, 5170048, 17651648, 60266496, 205762688, 702517760, 2398545664, 8189147136, 27959497216, 95459694592, 325919783936, 1112759746560, 3799199418368, 12971278180352, 44286713884672, 151204299177984
COMMENTS
Gives the number of L-convex polyominoes with n cells, that is convex polyominoes where any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientation of the L). - Simone Rinaldi (rinaldi(AT)unisi.it), Feb 19 2007
Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 2) is "size of raises in pot-limit poker, one blind, maximum raising".
Dimensions of the graded components of the Hopf algebra of noncommutative multi-symmetric functions of level 2. For level r, the sequence would be the INVERT transform of binomial(n+r-1,n). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
The sum of the numbers in the n-th row of the summatory Pascal triangle ( A059576). - Ron R. King, Jan 22 2009 (1 + 2x + 7x^2 + 24x^3 + ...) = 1 / (1 - 2x - 3x^2 - 4x^3 - ...). - Gary W. Adamson, Jul 27 2009 Let M be a triangle with the odd-indexed Fibonacci numbers (1, 2, 5, 13, ...) in every column, with the leftmost column shifted upwards one row. A003480 = lim_{n->oo} M^n, the left-shifted vector considered as a sequence. The analogous operation using the even-indexed Fibonacci numbers generates A001835 starting with offset 1. - Gary W. Adamson, Jul 27 2010 a(n) is the number of generalized compositions of n when there are i+1 different types of the part i, (i=1,2,...). - Milan Janjic, Sep 24 2010 Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t + 3*t^2 + 4*t^3 + ...)),
A001003(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. - Tom Copeland, Sep 06 2011
REFERENCES
G. Castiglione and A. Restivo, L-convex polyominoes: a survey, Chapter 2 of K. G. Subranian et al., eds., Formal Models, Languages and Applications, World Scientific, 2015.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
FORMULA
a(n) = (n+1)*a(0) + n*a(1) + ... + 3*a(n-2) + 2*a(n-1). - Amarnath Murthy, Aug 17 2002 G.f.: (1-x)^2/(1-4*x+2*x^2). - Simon Plouffe in his 1992 dissertation G.f.: 1/( 1 - Sum_{k>=1} (k+1)*x^k ).
a(n+1)*a(n+1) - a(n+2)*a(n) = 2^n, n > 0. - D. G. Rogers, Jul 12 2004
For n > 0, a(n) = ((2+sqrt(2))^(n+1) - (2-sqrt(2))^(n+1))/(4*sqrt(2)). - Rolf Pleisch, Aug 03 2009 If the leading 1 is removed, 2, 7, 24, ... is the binomial transform of 2, 5, 12, 29, ..., which is A000129 without its first 2 terms, and the second binomial transform of 2, 3, 4, 6, ..., which is A029744, again without its leading 1. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009 a(n) = Sum((1+p_1)*(1+p_2)*...*(1+p_m)), summation being over all compositions (p_1, p_2, ..., p_m) of n. Example: a(3)=24; indeed, the compositions of 3 are (1,1,1), (1,2), (2,1), (3) and we have 2*2*2 + 2*3 + 3*2 + 4 = 24. - Emeric Deutsch, Oct 17 2010 a(n) = Sum_{k>=0} binomial(n+2*k-1,n) / 2^(k+1). - Vaclav Kotesovec, Dec 31 2013 E.g.f.: (1 + exp(2*x)*(cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)))/2. - Stefano Spezia, May 20 2024
MAPLE
INVERT([seq(n+1, n=1..20)]); # Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
MATHEMATICA
a[0]=1; a[1]=2; a[2]=7; a[n_]:=a[n]=4*a[n-1] - 2*a[n-2]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 22 2011 *) Join[{1}, LinearRecurrence[{4, -2}, {2, 7}, 40]] (* Harvey P. Dale, Oct 23 2011 *)
PROG
(PARI) a(n)=polcoeff((1-x)^2/(1-4*x+2*x^2)+x*O(x^n), n)
(PARI) a(n)=local(x); if(n<1, n==0, x=(2+quadgen(8))^n; imag(x)+real(x)/2)
(Haskell)
a003480 n = a003480_list !! n
a003480_list = 1 : 2 : 7 : (tail $ zipWith (-)
(tail $ map (* 4) a003480_list) (map (* 2) a003480_list))
a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.
+10
37
0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264
COMMENTS
For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022
FORMULA
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022 a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022
MATHEMATICA
LinearRecurrence[{2, -10}, {0, 1}, 50]
PROG
(Magma) I:=[0, 1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011 (SageMath) [lucas_number1(n, 2, 10) for n in (0..50)] # G. C. Greubel, Jun 10 2022
CROSSREFS
Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=0, a(1)=1.
+10
29
0, 1, 4, 18, 80, 356, 1584, 7048, 31360, 139536, 620864, 2762528, 12291840, 54692416, 243353344, 1082798208, 4817899520, 21437194496, 95384577024, 424412697088, 1888419942400, 8402505163776, 37386860539904, 166352452487168
COMMENTS
Lower left term in matrix powers of [(1,5); (1,3)]. Convolved with (1, 2, 0, 0, 0, ...) the result is A164549: (1, 6, 26, 116, ...). - Gary W. Adamson, Aug 10 2016 For n>0, a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017
FORMULA
G.f.: x/(1 - 4*x - 2*x^2).
a(n) = (-i*sqrt(2))^(n-1) U(n-1, i*sqrt(2)) where U is the Chebyshev polynomial of the second kind and i^2 = -1.
a(n) = ((2+sqrt(6))^n - (2-sqrt(6))^n)/(2 sqrt(6)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009, Jan 07 2009
E.g.f.: sinh(sqrt(6)*x)*exp(2*x)/sqrt(6).
Number of zeros in substitution system {0 -> 11, 1 -> 11011} at step n from initial string "1" (1 -> 11011 -> 1101111011111101111011 -> ...). (End)
MATHEMATICA
a[n_Integer] := (-I Sqrt[2])^(n - 1) ChebyshevU[ n - 1, I Sqrt[2] ]
a[n_]:=(MatrixPower[{{1, 5}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[Abs[a[n]], {n, -1, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *) t={0, 1}; Do[AppendTo[t, 4*t[[-1]]+2*t[[-2]]], {n, 2, 23}]; t (* or *) LinearRecurrence[{4, 2}, {0, 1}, 24] (* Indranil Ghosh, Feb 21 2017 *)
PROG
(Sage) [lucas_number1(n, 4, -2) for n in range(0, 23)] # Zerinvary Lajos, Apr 23 2009 (Magma) I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 12 2011
a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 2.
+10
27
1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152
COMMENTS
a(n+1) is the number of palindromic words of length n using a two-letter alphabet. - Michael Somos, Mar 20 2011
FORMULA
a(n) = 2^((1/4)*(2*n - 1 + (-1)^n)).
G.f.: x*(1 + 2*x)/(1 - 2*x^2).
G.f.: x / (1 - 2*x / (1 + x / (1 + x))) = x * (1 + 2*x / (1 - x / (1 - x / (1 + 2*x)))). - Michael Somos, Jan 03 2013 E.g.f.: cosh(sqrt(2)*x) + sinh(sqrt(2)*x)/sqrt(2) - 1. - Stefano Spezia, Feb 05 2023
EXAMPLE
x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + 16*x^8 + 16*x^9 + 32*x^10 + ...
MATHEMATICA
LinearRecurrence[{0, 2}, {1, 2}, 50] (* Paolo Xausa, Feb 02 2024 *)
PROG
(Magma) [ n le 2 select n else 2*Self(n-2): n in [1..43] ];
(PARI) {a(n) = if( n<1, 0, 2^(n\2))} /* Michael Somos, Mar 20 2011 */ (Sage)
x, y = 1, 1
while True:
yield x
x, y = x + y, x - y
CROSSREFS
Binomial transform is A078057, second binomial transform is A007070, third binomial transform is A102285, fourth binomial transform is A163350, fifth binomial transform is A163346. The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022
Power ceiling-floor sequence of (golden ratio)^4.
+10
20
7, 47, 323, 2213, 15169, 103969, 712615, 4884335, 33477731, 229459781, 1572740737, 10779725377, 73885336903, 506417632943, 3471038093699, 23790849022949, 163064905066945, 1117663486445665, 7660579500052711
COMMENTS
Let f = floor and c = ceiling. For x > 1, define four sequences as functions of x, as follows:
p1(0) = f(x), p1(n) = f(x*p1(n-1));
p2(0) = f(x), p2(n) = c(x*p2(n-1)) if n is odd and p2(n) = f(x*p1(n-1)) if n is even;
p3(0) = c(x), p3(n) = f(x*p3(n-1)) if n is odd and p3(n) = c(x*p3(n-1)) if n is even;
p4(0) = c(x), p4(n) = c(x*p4(n-1)).
The present sequence is given by a(n) = p3(n).
Following the terminology at A214986, call the four sequences power floor, power floor-ceiling, power ceiling-floor, and power ceiling sequences. In the table below, a sequence is identified with an A-numbered sequence if they appear to agree except possibly for initial terms. Notation: S(t)=sqrt(t), r = (1+S(5))/2 = golden ratio, and Limit = limit of p3(n)/p2(n). x ......p1..... p2..... p3..... p4.......Limit
...
Properties of p1, p2, p3, p4:
(1) If x > 2, the terms of p2 and p3 interlace: p2(0) < p3(0) < p2(1) < p3(1) < p2(2) < p3(2)... Also, p1(n) <= p2(n) <= p3(n) <= p4(n) <= p1(n+1) for all x>0 and n>=0.
(2) If x > 2, the limits L(x) = limit(p/x^n) exist for the four functions p(x), and L1(x) <= L2(x) <= L3(x) <= L4 (x). See the Mathematica programs for plots of the four functions; one of them also occurs in the Odlyzko and Wilf article, along with a discussion of the special case x = 3/2.
(3) Suppose that x = u + sqrt(v) where v is a nonsquare positive integer. If u = f(x) or u = c(x), then p1, p2, p3, p4 are linear recurrence sequences. Is this true for sequences p1, p2, p3, p4 obtained from x = (u + sqrt(v))^q for every positive integer q?
(4) Suppose that x is a Pisot-Vijayaraghavan number. Must p1, p2, p3, p4 then be linearly recurrent? If x is also a quadratic irrational b + c*sqrt(d), must the four limits L(x) be in the field Q(sqrt(d))?
(5) The Odlyzko and Wilf article (page 239) raises three interesting questions about the power ceiling function; it appears that they remain open.
FORMULA
a(n) = floor(r*a(n-1)) if n is odd and a(n) = ceiling(r*a(n-1)) if n is even, where a(0) = ceiling(r), r = (golden ratio)^4 = (7 + sqrt(5))/2.
a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3).
G.f.: (7 + 5*x - x^2)/((1 + x)*(1 - 7*x + x^2)).
a(n) = (10*(-2)^n+(10+3*sqrt(5))*(7-3*sqrt(5))^(n+2)+(10-3*sqrt(5))*(7+3*sqrt(5))^(n+2))/(90*2^n). - Bruno Berselli, Nov 14 2012 E.g.f.: exp(-x)*(5 + 2*exp(9*x/2)*(155*cosh(3*sqrt(5)*x/2) + 69*sqrt(5)*sinh(3*sqrt(5)*x/2)))/45. - Stefano Spezia, Oct 28 2024
EXAMPLE
a(0) = ceiling(r) = 7, where r = ((1+sqrt(5))/2)^4 = 6.8...; a(1) = floor(7*r) = 47; a(2) = ceiling(47) = 323.
MATHEMATICA
(* Program 1. A214992 and related sequences *) x = GoldenRatio^4; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}] (* A049685 *) Table[p2[n], {n, 0, z}] (* A157335 *) Table[p3[n], {n, 0, z}] (* A214992 *) Table[p4[n], {n, 0, z}] (* A004187 *) Table[p4[n] - p1[n], {n, 0, z}] (* A004187 *) Table[p3[n] - p2[n], {n, 0, z}] (* A098305 *) (* Program 2. Plot of power floor and power ceiling functions, p1(x) and p4(x) *)
f[x_] := f[x] = Floor[x]; c[x_] := c[x] = Ceiling[x];
p1[x_, 0] := f[x]; p1[x_, n_] := f[x*p1[x, n - 1]];
p4[x_, 0] := c[x]; p4[x_, n_] := c[x*p4[x, n - 1]];
Plot[Evaluate[{p1[x, 10]/x^10, p4[x, 10]/x^10}], {x, 2, 3}, PlotRange -> {0, 4}]
(* Program 3. Plot of power floor-ceiling and power ceiling-floor functions, p2(x) and p3(x) *)
f[x_] := f[x] = Floor[x]; c[x_] := c[x] = Ceiling[x];
p2[x_, 0] := f[x]; p3[x_, 0] := c[x];
p2[x_, n_] := If[Mod[n, 2] == 1, c[x*p2[x, n - 1]], f[x*p2[x, n - 1]]]
p3[x_, n_] := If[Mod[n, 2] == 1, f[x*p3[x, n - 1]], c[x*p3[x, n - 1]]]
Plot[Evaluate[{p2[x, 10]/x^10, p3[x, 10]/x^10}], {x, 2, 3}, PlotRange -> {0, 4}]
Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 2x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.
+10
19
1, 2, 2, 5, 6, 3, 12, 20, 12, 4, 29, 60, 50, 20, 5, 70, 174, 180, 100, 30, 6, 169, 490, 609, 420, 175, 42, 7, 408, 1352, 1960, 1624, 840, 280, 56, 8, 985, 3672, 6084, 5880, 3654, 1512, 420, 72, 9, 2378, 9850, 18360, 20280, 14700, 7308, 2520, 600, 90, 10
COMMENTS
Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.
FORMULA
p(n, x) = u*p(n-1, x) + v*p(n-2, x) for n >= 3, where p(1, x) = 1, p(2, x) = 2 + 2 x, u = p(2, x), and v = 1 - 2x - x^2.
p(n, x) = k*(b^n - c^n), where k = sqrt(1/8), b = x + 1 - sqrt(2), c = x + 1 + sqrt(2).
The row polynomials p(n, x) = Sum_{k=0..n-1} T(n, k) * x^k satisfy the equation p'(n, x) = n * p(n-1, x) where p' is the first derivative of p.
T(n, k) = T(n-1, k-1) * n / k for 0 < k < n and T(n, 0) = A000129(n) for n > 0. T(n, k) = A000129(n-k) * binomial(n, k) for 0 <= k < n. G.f.: t / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)
EXAMPLE
First nine rows:
[n\k] 0 1 2 3 4 5 6 7 8
[1] 1;
[2] 2 2;
[3] 5 6 3;
[4] 12 20 12 4;
[5] 29 60 50 20 5;
[6] 70 174 180 100 30 6;
[7] 169 490 609 420 175 42 7;
[8] 408 1352 1960 1624 840 280 56 8;
[9] 985 3672 6084 5880 3654 1512 420 72 9;
.
Row 4 represents the polynomial p(4,x) = 12 + 20 x + 12 x^2 + 4 x^3, so that (T(4,k)) = (12, 20, 12, 4), k = 0..3.
MAPLE
P := proc(n) option remember; ifelse(n <= 1, n, 2*P(n - 1) + P(n - 2)) end:
T := (n, k) -> P(n - k) * binomial(n, k):
for n from 1 to 9 do [n], seq(T(n, k), k = 0..n-1) od;
MATHEMATICA
p[1, x_] := 1; p[2, x_] := 2 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
CROSSREFS
Cf. A000129 (column 1), A361732 (column 2), A000027 (T(n,n-1)), A007070 (row sums, p(n,1)), A077957 (alternating row sums, p(n,-1)), A081179 (p(n,2), A077985 (p(n,-2), A081180 (p(n,3)), A007070 (p(n,-3)), A081182 (p(n,4)), A094440, A367208, A367209, A367210.
3rd binomial transform of (0,1,0,2,0,4,0,8,0,16,...).
+10
17
0, 1, 6, 29, 132, 589, 2610, 11537, 50952, 224953, 993054, 4383653, 19350540, 85417669, 377052234, 1664389721, 7346972688, 32431108081, 143157839670, 631929281453, 2789470811028, 12313319895997, 54353623698786
COMMENTS
Binomial transform of 0, 1, 4, 14, 48, ... ( A007070 with offset 1) and second binomial transform of A000129. - R. J. Mathar, Dec 10 2011
FORMULA
a(n) = 6*a(n-1) - 7*a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1-6*x+7*x^2).
a(n) = ((3+sqrt(2))^n - (3-sqrt(2))^n)/(2*sqrt(2)). [Corrected by Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008]
a(n) = 3^(n-1) Sum_{i>=0} binomial(n, 2i+1) * (2/9)^i. - Sergio Falcon, Mar 15 2016 a(n) = 2^(-1/2)*7^(n/2)*sinh(n*arcsinh(sqrt(2/7))). - Robert Israel, Mar 15 2016 a(n) = 7^((n-1)/2)*ChebyshevU(n-1, 3/sqrt(7)). - G. C. Greubel, Jan 14 2024
MAPLE
f:= gfun:-rectoproc({a(n) = 6*a(n-1)-7*a(n-2), a(0)=0, a(1)=1}, a(n), remember):
MATHEMATICA
CoefficientList[Series[x/(1-6 x +7 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *) LinearRecurrence[{6, -7}, {0, 1}, 41] (* G. C. Greubel, Jan 14 2024 *)
PROG
(Sage) [lucas_number1(n, 6, 7) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009 (Magma) I:=[0, 1]; [n le 2 select I[n] else 6*Self(n-1)-7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013
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