Displaying 1-10 of 11 results found.
1, 1, 3, 1, 5, 3, 7, 2, 9, 5, 11, 3, 13, 7, 15, 4, 17, 9, 19, 5, 21, 11, 23, 6, 25, 13, 27, 7, 29, 15, 31, 8, 33, 17, 35, 9, 37, 19, 39, 10, 41, 21, 43, 11, 45, 23, 47, 12, 49, 25, 51, 13, 53, 27, 55, 14, 57, 29, 59, 15, 61, 31, 63, 16, 65, 33, 67, 17, 69, 35, 71, 18, 73, 37, 75, 19
COMMENTS
We make some general remarks about the sequence a(n) = numerator(n/(n + k)) = n/gcd(n,k) for k a fixed positive integer. The present sequence is the case k = 4. Several other cases are listed in the Crossrefs. In addition to being multiplicative these sequences are also strong divisibility sequences, that is, gcd(a(n),a(m)) = a(gcd(n, m)) for n, m >= 1. In particular, it follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m).
By the multiplicativeness and strong divisibility property of the sequence a(n) it follows that if gcd(n, m) = 1 then a(a(n)*a(m) ) = a(a(n)) * a(a(m)), a(a(a(n))*a(a(m)) ) = a(a(a(n))) * a(a(a(m))) and so on.
The sequence a(n) has the rational generating function Sum_{d divides k} f(d)*x^d/(1 - x^d)^2, where f(n) is the Dirichlet inverse of the Euler totient function A000010. f(n) is a multiplicative function defined on prime powers p^k by f(p^k) = 1 - p. See A023900. Cf. A181318. (End)
FORMULA
G.f.: x*(1 +x +3*x^2 +x^3 +3*x^4 +x^5 +x^6)/(1 - x^4)^2.
a(n) = 2*a(n-4) - a(n-8).
a(n) = (n/16)*(11 - 5*(-1)^n - i^n - (-i)^n). - Ralf Stephan, Mar 15 2003 a(2*n+1) = a(4*n+2) = 2*n+1, a(4*n+4) = n+1. - Ralf Stephan, Jun 10 2005 Multiplicative with a(2^e) = 2^max(0, e-2), a(p^e) = p^e, p >= 3. - Mitch Harris, Jun 29 2005 Dirichlet g.f.: zeta(s-1)*(1-1/2^s-1/2^(2s)). (End)
a(n) = numerator(Sum_{k=1..n} 1/((k+1)*(k+2))). This summation has a closed form of 1/2 - 1/(n+2) and denominator of A145979(n). - Gary Detlefs, Sep 16 2011 a((2*n-1)*2^p) = ceiling(2^(p-2))*(2*n-1), p >= 0 and n >= 1. - Johannes W. Meijer, Feb 06 2013 O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - F(x^2) - F(x^4), where F(x) = x/(1 - x)^2.
More generally, Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) + (1 - 2^m)*( F(m,x^2) + F(m,x^4) ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292. Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence a(n) produces generating functions for the sequences ((n^m)*a(n))n>=1 for m in Z. Some examples are given below.
(End)
Sum_{k=1..n} a(k) ~ (11/32) * n^2. - Amiram Eldar, Nov 25 2022 E.g.f.: x*(8*cosh(x) + sin(x) + 3*sinh(x))/8. - Stefano Spezia, Dec 02 2023
EXAMPLE
Sum_{n >= 1} n*a(n)*x^n = G(x) - 2*G(x^2) - 4*G(x^4), where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (1/2)*H(x^2) - (1/4)*H(x^4), where H(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (1/2^2)*L(x^2) - (1/4^2)*L(x^4), where L(x) = Log(1/(1 - x)).
Sum_{n >= 1} (1/a(n))*x^n = L(x) + (1/2)*L(x^2) + (1/2)*L(x^4). (End)
MATHEMATICA
f[n_]:= n/GCD[n, 4]; Array[f, 80]
PROG
(PARI) { for (n=1, 1000, write("b060819.txt", n, " ", n / gcd(n, 4)); ) } \\ Harry J. Smith, Jul 12 2009 (Haskell)
(Magma) [n/GCD(n, 4): n in [1..80]]; // G. C. Greubel, Sep 19 2018
a(4*n) = n, a(1+2*n) = 4+8*n, a(2+4*n) = 2+4*n.
+10
6
0, 4, 2, 12, 1, 20, 6, 28, 2, 36, 10, 44, 3, 52, 14, 60, 4, 68, 18, 76, 5, 84, 22, 92, 6, 100, 26, 108, 7, 116, 30, 124, 8, 132, 34, 140, 9, 148, 38, 156, 10, 164, 42, 172, 11, 180, 46, 188, 12, 196, 50, 204, 13, 212, 54, 220, 14, 228, 58, 236, 15, 244, 62
FORMULA
a(n) = 2*a(n-4) - a(n-8) for n>7.
a(4*n) = a(-4+4*n) + 1.
a(1+4*n) = a(-3+4*n) + 16.
a(2+4*n) = a(-2+4*n) + 4.
a(3+4*n) = a(-1+4*n) + 16. See A177499. G.f.: x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2.
a(n) = (64-3*(1+(-1)^n)*(9+i^n))*n/16 with i=sqrt(-1).
a(n)/a(n-4) = n/(n-4) for n>4. (End)
Sum_{k=1..n} a(k) ~ (37/32)*n^2. - Amiram Eldar, Oct 07 2023
MATHEMATICA
Table[8 n/(11 + 9 Cos[Pi*n] + 12 Cos[n*Pi/2]), {n, 0, 80}] (* Wesley Ivan Hurt, Jul 06 2016 *) CoefficientList[Series[x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2, {x, 0, 50}], x] (* G. C. Greubel, Sep 20 2018 *) LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, -1}, {0, 4, 2, 12, 1, 20, 6, 28}, 70] (* Harvey P. Dale, Aug 14 2019 *)
PROG
(Magma) [(64-3*(1+(-1)^n)*(9+(-1)^(n div 2)))*n/16 : n in [0..80]]; // Wesley Ivan Hurt, Jul 06 2016 (PARI) x='x+O('x^50); concat([0], Vec(x*(4+2*x+12*x^2+x^3+12*x^4+ 2*x^5 +4*x^6)/(1-x^4)^2)) \\ G. C. Greubel, Sep 20 2018
a(4*n) = n*(16*n^2-1)/3, a(2*n+1) = n*(n+1)*(2*n+1)/6, a(4*n+2) = (4*n+1)*(4*n+2)*(4*n+3)/6.
+10
5
0, 0, 1, 1, 5, 5, 35, 14, 42, 30, 165, 55, 143, 91, 455, 140, 340, 204, 969, 285, 665, 385, 1771, 506, 1150, 650, 2925, 819, 1827, 1015, 4495, 1240, 2728, 1496, 6545, 1785, 3885, 2109, 9139, 2470, 5330, 2870, 12341, 3311, 7095, 3795, 16215, 4324
COMMENTS
a(n+2) is divisible by A060819(floor(n/3)). a(n) is divisible by A176672(floor(n/3)). Denominator of a(n)/n is of period 24: 1,1,3,4,1,6,1,4,3,1,1,12,1,2,3,4,1,3,1,4,3,2,1,12 (two successive palindromes).
This is the fifth column of the triangle A107711, hence the formula involving gcd(n+2,4) given below follows. - Wolfdieter Lang, Feb 24 2014
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).
FORMULA
a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).
a(n)/a(n+4) = n*(n^2-1)/((n+3)*(n+4)*(n+5)).
a(n)/a(n+12) = (n-1)*n*(n+1)/((n+11)*(n+12)*(n+13)).
G.f.: (x^2 + x^3 + 5*x^4 + 5*x^5 + 31*x^6 + 10*x^7 + 22*x^8 + 10*x^9 + 31*x^10 + 5*x^11 + 5*x^12 + x^13 + x^14) / ((1-x)^4*(1+x)^4*(1 + 4*x^2 + 6*x^4 + 4*x^6 + x^8)). - R. J. Mathar, Mar 10 2012 G.f.: (1 + x^12 + x*(1+x^10) + 5*x^2*(1+x^8) + 5*x^3*(1+x^7) + 31*x^4*(1+x^4) + 10*x^5*(1+x^2) + 22*x^6)/(1-x^4)^4. This is the preceding g.f. rewritten.
a(n) = binomial(n+1,3)*gcd(n+2,4)/4, n >= 0. From the g.f., see a comment above on A107711. (End) a(n) = (n*(n-1)*((n+1)*(4+2*(-1)^n + (1+(-1)^n)*(-1)^((2*n+3+(-1)^n)/4))))/48. - Luce ETIENNE, Jan 01 2015 Sum_{n>=2} 1/a(n) = 12 - 27*log(2)/2. - Amiram Eldar, Aug 12 2022
MATHEMATICA
CoefficientList[Series[(x^2 + x^3 + 5 x^4 + 5 x^5 + 31 x^6 + 10 x^7 + 22 x^8 + 10 x^9 + 31 x^10 + 5 x^11 + 5 x^12 + x^13 + x^14)/((1 - x)^4 (1 + x)^4 (1 + 4 x^2 + 6 x^4 + 4 x^6 + x^8)), {x, 0, 47}], x] (* Bruno Berselli, Mar 11 2012 *)
PROG
[a, npr] ,
if equal(mod(n, 4), 0) then (
a : n/12*(n^2-1)
) else if equal(mod(n, 2), 0) then (
a : (n-1)*n*(n+1)/6
) else (
npr : (n-1)/2,
a : npr*(npr+1)*n/6
) ,
return(a)
(PARI) vector(50, n, n--; binomial(n+1, 3)*gcd(n+2, 4)/4) \\ G. C. Greubel, Sep 20 2018 (Magma) [Binomial(n+1, 3)*GCD(n+2, 4)/4: n in [0..50]]; // G. C. Greubel, Sep 20 2018
CROSSREFS
Cf. A000034, A000292, A002415, A051724, A060819, A061037, A107711, A138190, A145979, A176672, A176895.
Period 4: repeat [1, 16, 4, 16].
+10
4
1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16
COMMENTS
Continued fraction expansion of (44+sqrt(2442))/88. (End)
FORMULA
a(n) = a(n-4) for n > 3.
G.f.: (1+16*x+4*x^2+16*x^3)/(1-x^4). (End)
a(n) = (37 - 6*cos(n*Pi/2) - 27*cos(n*Pi) - 27*I*sin(n*Pi))/4. - Wesley Ivan Hurt, Jul 09 2016
Denominators of the Inverse semi-binomial transform of A001477(n) read downwards antidiagonals.
+10
4
1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 2, 2, 8, 2, 1, 1, 4, 1, 16, 16, 1, 2, 1, 8, 8, 32, 16, 1, 1, 4, 4, 16, 8, 64, 64, 1, 2, 2, 8, 4, 32, 32, 128, 16, 1, 1, 4, 2, 16, 16, 64, 8, 256, 256, 1, 2, 1, 8, 8, 32, 4, 128, 128, 512, 256, 1, 1, 4, 4, 16, 2, 64, 64, 256, 128, 1024, 1024
COMMENTS
Starting from any sequence a(k) in the first row, define the array T(n,k) of the inverse semi-binomial transform by T(0,k) = a(k), T(n,k) = T(n-1,k+1) -T(n-1,k)/2, n>=1.
Here, where the first row is the nonnegative integers, the array is
1/2 9/16 5/8 11/16 3/4 13/16 7/8 15/16 1 = A106617(n+8)/TBD 5/16 11/32 3/8 13/32 7/16 15/32 1/2 17/32 9/16
3/16 13/64 7/32 15/64 1/4 17/64 9/32 19/64 5/16
7/64 15/128 1/8 17/128 9/64 19/128 5/32 21/128 11/64
1/16 17/256 9/128 19/256 5/64 21/256 11/128 23/256 3/32.
The first column contains 0, followed by fractions A000265/ A084623, that is Oresme numbers n/2^n multiplied by 2 (see A209308).
EXAMPLE
The array of denominators starts:
1 1 1 1 1 1 1 1 1 1 1 ...
1 2 1 2 1 2 1 2 1 2 1 ...
1 4 2 4 1 4 2 4 1 4 2 ...
4 8 1 8 4 8 2 8 4 8 1 ...
2 16 8 16 4 16 8 16 1 16 8 ...
16 32 8 32 16 32 2 32 16 32 8 ...
16 64 32 64 4 64 32 64 16 64 32 ...
64 128 8 128 64 128 32 128 64 128 16 ...
16 256 128 256 64 256 128 256 32 256 128 ...
256 512 128 512 256 512 64 512 256 512 128 ...
All entries are powers of 2.
MAPLE
A213268frac := proc(n, k)
if n = 0 then
return k ;
else
return procname(n-1, k+1)-procname(n-1, k)/2 ;
end if;
end proc:
denom(A213268frac(n, k)) ;
MATHEMATICA
T[0, k_] := k; T[n_, k_] := T[n, k] = T[n-1, k+1] - T[n-1, k]/2; Table[T[n-k, k] // Denominator, {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 12 2014 *)
a(4*n) = n*(4*n-1); a(2*n+1) = n*(n+1)/2; a(4*n+2) = (2*n+1)*(4*n+1).
+10
2
0, 0, 1, 1, 3, 3, 15, 6, 14, 10, 45, 15, 33, 21, 91, 28, 60, 36, 153, 45, 95, 55, 231, 66, 138, 78, 325, 91, 189, 105, 435, 120, 248, 136, 561, 153, 315, 171, 703, 190, 390, 210, 861, 231, 473, 253, 1035, 276, 564, 300, 1225, 325
COMMENTS
a(n) is divisible by the n-th term of the sequence 3, 3, 1, 1, 3, 3 (periodically repeated with period 6).
a(n) is divisible by b(floor((n-1)/3)), where b(n) = 1, 3, 2, 3, 7, 3, 5, 3, 13, 3, 8, 3, 19, 3,... , n>=0, is defined by inserting a 3 after each entry of A165355. (n+1)*(n+2)*(n+3)/2=3* A000292(n+1) is divisible by a(n+2), so there is an integer sequence c(n)= 3* A000292(n+1)/a(n+2) = 3, 12, 10, 20, 7, 28, 18,... with c(2*n)= A123167(n+1) and c(n)/ A109613(n+2)= A176895(n). The sequence of denominators of a(n+2)/n has period length 8: 1, 2, 1, 4, 1, 1, 1, 4.
A table T(k,c) = a(1+c*(1+2k)) of (2*k+1)-sections starts as follows:
0 1 1 3 3 15...
0 3 6 45 21 60...
0 15 15 60 55 325...
0 14 28 231 105 315...
0 45 45 189 171 1035...
The table of T'(k,c) = T(k,c)/(2k+1), columns c>=0, looks as follows, construction similar to A165943: 0 1 1 3 3 15 6 14 k=0
0 1 2 15 7 20 15 77 k=1
0 3 3 12 11 65 24 63 k=2
0 2 4 33 15 45 33 175 k=3
0 5 5 21 19 115 42 112 k=4
0 3 6 51 23 70 51 273 k=5
The entries T'(k,c) are divisible by A060819(c). Differences are T'(2,c)-T'(0,c) = T'(4,c)-T'(2,c) = 0, 2, 2, 9, 8, 50, 18, 49, 32, ... which is A168077(c) multiplied by the c-th term of the period-4 sequence 2, 2, 2, 1. Differences are T'(3,c)- T'(1,c) = T'(5,c)-T'(3,c) = 0, 1, 2, 18, 8, 25, 18, 98, 32,... which is A168077(c) multiplied by the period-4 sequence 2, 1, 2, 2. The reduced fractions T'(0,c)/T'(1,c) = 1, 1/2, 1/5, 3/7, 3/4, 2/5, 2/11, 5/13, 5/7, 3/8, 3/17, 7/19, .., c>=1, have a numerator sequence A026741(floor(c/2)+1). The denominator sequence is f(c) = 1, 2, 5, 7, 4, 5,.. = A001651(c+1)/ A130658(c+1), with f(2*c+1) +f(2*c+2) = 3, 12, 9, 24 .. =3* A022998(c).
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).
FORMULA
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
G.f.: -x^2*(3*x^8+x^7+5*x^6+3*x^5+12*x^4+3*x^3+3*x^2+x+1)/ ((x-1)^3*(x+1)^3*(x^2+1)^3). - R. J. Mathar, Mar 22 2012 a(n) = (4*n^2-3*n-1+(2*n^2-3*n+1)*(-1)^n + n*(n-1)*(1+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4))/16. - Luce ETIENNE, May 13 2016 Sum_{n>=2} 1/a(n) = 2 - Pi/4 + 7*log(2)/2. - Amiram Eldar, Aug 12 2022
MAPLE
if n mod 4 = 0 then
return n/4*(n-1) ;
elif n mod 2 = 1 then
return (n-1)*(n+1)/8 ;
else
return (n-1)*n/2 ;
end if;
MATHEMATICA
Clear[b]; b[1] = 0; b[2] = 0; b[3] = 1; b[4] = 1; b[5] = 3; b[6] = 3; b[7] = 15; b[8] = 6; b[n_Integer] := b[n] = ((-2 + n) (-4 (-4 + n) (-3 + n) (-2 + n) (8 + n (-9 + 2 n)) b[-3 + n] + (-5 + n) ((-3 +n) ((-4 + n) (211 + 2 n (-215 + n (147 + n (-41 + 4 n)))) - 4 (-1 + n) (19 + n (-13 + 2 n)) b[-2 + n]) - 4 (-4 + n)^2 (8 + n (-9 + 2 n)) b[-1 + n])))/(4 (-5 + n) (-4 + n) (-3 + n)^2 (19 + n (-13 + 2 n)))
LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 0, 1, 1, 3, 3, 15, 6, 14, 10, 45, 15}, 60] (* Harvey P. Dale, Nov 23 2015 *)
PROG
(PARI) x='x+O('x^50); concat([0, 0], Vec(-x^2*(3*x^8+x^7+5*x^6+3*x^5+12*x^4+3*x^3+3*x^2+x+1)/ ((x-1)^3*(x+1)^3*(x^2+1)^3))) \\ G. C. Greubel, Jun 23 2017
A permutation of essentially the duplicate nonnegative numbers: a(4n) = n + 1/2 - (-1)^n/2, a(2n+1) = a(4n+2) = 2n+1.
+10
2
0, 1, 1, 3, 2, 5, 3, 7, 2, 9, 5, 11, 4, 13, 7, 15, 4, 17, 9, 19, 6, 21, 11, 23, 6, 25, 13, 27, 8, 29, 15, 31, 8, 33, 17, 35, 10, 37, 19, 39, 10, 41, 21, 43, 12, 45, 23, 47, 12, 49, 25, 51, 14, 53, 27, 55, 14, 57, 29, 59, 16, 61, 31, 63, 16
COMMENTS
0 is at its own place. Distance between the two (2*k+1)'s: 2*k+1 terms. 0 is in position 0, the first 1 in position 1, the second 1 in position 2, the first 2 in position 4, the second 2 in position 8. Hence, r(n) = 0, 1, 2, 4, 8, 3, 6, 12, 16, 5, 10, 20, 24, ..., a permutation of A001477. See A225055. The recurrence r(n) = r(n-4) + r(n-8) - r(n-12) is the same as for a(n). A061037(n+3) is divisible by a(n+5) (= 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 3, ...). Hence a link, via A212831 and A214282, between the Catalan numbers A000108 and the Balmer series.
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,1,0,0,0,-1).
FORMULA
a(n) = 3*a(n-8) - 3*a(n-16) + a(n-24).
a(n+4) = a(n) + period 8: repeat [2, 4, 2, 4, 0, 4, 2, 4].
a(n+8) = a(n) + period 4: repeat [2, 8, 4, 8] (= 2 * A176895). a(n) = n*(1+floor((2-n)/4)+floor((n-2)/4))/2+n*(1+floor((1-n)/2)+floor((n-1)/2))+(-n-2+2*(-1)^(n/4))*(ceiling(n/4)-floor(n/4)-1)/4. - Wesley Ivan Hurt, Sep 14 2014 G.f.: x*(x^10+x^9+3*x^8+4*x^6+2*x^5+4*x^4+2*x^3+3*x^2+x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)). - Colin Barker, Sep 15 2014
MAPLE
A246416:=n->n*(1+floor((2-n)/4)+floor((n-2)/4))/2+n*(1+floor((1-n)/2)+floor((n-1)/2))+(-n-2+2*(-1)^(n/4))*(ceil(n/4)-floor(n/4)-1)/4: seq( A246416(n), n=0..50); # Wesley Ivan Hurt, Sep 14 2014
MATHEMATICA
Table[n (1 + Floor[(2 - n)/4] + Floor[(n - 2)/4])/2 + n (1 + Floor[(1 - n)/2] + Floor[(n - 1)/2]) + (-n - 2 + 2 (-1)^(n/4)) (Ceiling[n/4] - Floor[n/4] - 1)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Sep 14 2014 *) a[n_] := Switch[Mod[n, 4], 0, n/4-(-1)^(n/4)/2+1/2, 1|3, n, 2, n/2]; Table[a[n], {n, 0, 64}] (* Jean-François Alcover, Oct 09 2014 *) LinearRecurrence[{0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, -1}, {0, 1, 1, 3, 2, 5, 3, 7, 2, 9, 5, 11}, 70] (* Harvey P. Dale, Mar 23 2015 *)
PROG
(Magma) I:=[0, 1, 1, 3, 2, 5, 3, 7, 2, 9, 5, 11, 4, 13, 7, 15, 4, 17, 9, 19, 6, 21, 11, 23]; [n le 24 select I[n] else 3*Self(n-8)-3*Self(n-16)+Self(n-24): n in [1..80]]; // Vincenzo Librandi, Oct 15 2014
a(4n) = n + 1/2 - (-1)^n/2 + (-1)^n, a(2n+1) = 2*n + 5, a(4n+2) = 2*n + 3.
+10
2
1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 3, 17, 9, 19, 5, 21, 11, 23, 5, 25, 13, 27, 7, 29, 15, 31, 7, 33, 17, 35, 9, 37, 19, 39, 9, 41, 21, 43, 11, 45, 23, 47, 11, 49, 25, 51, 13, 53, 27, 55, 13, 57, 29, 59, 15, 61, 31, 63, 15, 65, 33, 67
COMMENTS
Essentially a permutation of A129756 (odd numbers repeated four times). a(-1) = 3, a(-2) = a(-3) = 1.
Distance between the first two (2*k+1)'s: 2*k+1 terms. Distance between the last two (2*n+1)'s: 4 terms. Essentially same distances as in -a(-n) = -1, -3, -1, -1, 1, 1, 1, 3, 1, 5, 3, 7, 3, 9, 5, 11, 3, 13, 7, 15, 5, 17, 9, 19, 5, 21, 11, 23, 7, 25, 13, 27, 7, ... .
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,1,0,0,0,-1).
FORMULA
a(n) = a(n-4) + a(n-8) - a(n-12).
A246416(n+4) - a(n) = sequence of period 4: [1, 0, 0, 0].
a(n+4) - a(n) = sequence of period 8: [0, 4, 2, 4, 2, 4, 2, 4].
G.f.: -(3*x^11+x^10+x^9-x^8-4*x^7-2*x^6-4*x^5-7*x^3-3*x^2-5*x-1) / ((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)). - Colin Barker, Sep 21 2014 a(n) = a(n-8) + sequence of period 4: [2, 8, 4, 8] (= 2* A176895(n)). a(n) = (n+4)*(1-ceiling((2-n)/4)-ceiling((n-2)/4))/2+(n+4)*(1+floor((1-n)/2)+floor((n-1)/2))-(n+2+2(-1)^(n/4))*(ceiling(n/4)-floor(n/4)-1)/4. - Wesley Ivan Hurt, Sep 21 2014
MAPLE
A247617:=n->(n+4)*(1-ceil((2-n)/4)-ceil((n-2)/4))/2+(n+4)*(1+floor((1-n)/2)+floor((n-1)/2))-(n+2+2*(-1)^(n/4))*(ceil(n/4)-floor(n/4)-1)/4: seq( A247617(n), n=0..50); # Wesley Ivan Hurt, Sep 21 2014
MATHEMATICA
Table[(n + 4) (1 - Ceiling[(2 - n)/4] - Ceiling[(n - 2)/4])/2 + (n + 4) (1 + Floor[(1 - n)/2] + Floor[(n - 1)/2]) - (n + 2 + 2 (-1)^(n/4)) (Ceiling[n/4] - Floor[n/4] - 1)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Sep 21 2014 *)
PROG
(PARI) Vec(-(3*x^11+x^10+x^9-x^8-4*x^7-2*x^6-4*x^5-7*x^3-3*x^2-5*x-1)/((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)) + O(x^100)) \\ Colin Barker, Sep 21 2014 (Magma) I:=[1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15]; [n le 12 select I[n] else Self(n-4)+Self(n-8)-Self(n-12): n in [1..80]]; // Vincenzo Librandi, Oct 15 2014
a(4*n) = n-1. a(2*n+1) = a(4*n+2) = 2*n+1.
+10
1
-1, 1, 1, 3, 0, 5, 3, 7, 1, 9, 5, 11, 2, 13, 7, 15, 3, 17, 9, 19, 4, 21, 11, 23, 5, 25, 13, 27, 6, 29, 15, 31, 7, 33, 17, 35, 8, 37, 19, 39, 9, 41, 21, 43, 10, 45, 23, 47, 11, 49, 25, 51, 12, 53, 27, 55, 13, 57, 29, 59, 14, 61, 31, 63, 15, 65, 33, 67, 16, 69
COMMENTS
Consider the family of sequences with recurrence a(n) = 2*a(n-4)-a(n-8) where a(0) and a(4) move up in steps of 1. This here is characterized by a(0)=-1, a(4)=0:
-2, 1, 1, 3, -1, 5, 3, 7, 0, 9, 5, 11,...
-1, 1, 1, 3, 0, 5, 3, 7, 1, 9, 5, 11,... = a(n)
0, 1, 1, 3, 1, 5, 3, 7, 2, 9, 5, 11,... = A0608191, 1, 1, 3, 2, 5, 3, 7, 3, 9, 5, 11,... = b(n)
2, 1, 1, 3, 3, 5, 3, 7, 4, 9, 5, 11,... .
a(n+4)-b(n) = repeat -1, 4, 2, 4 with period of length 4.
FORMULA
a(n) = 2*a(n-4) - a(n-8).
G.f.: (-1+x+x^2+3*x^3+3*x^5+x^6+x^7+2*x^4)/((-1+x)^2*(1+x)^2*(x^2+1)^2). - R. J. Mathar, Apr 28 2013
MATHEMATICA
a[n_] := 1/16*(11*n-(-1)^n*(5*n+4)-2*(n+4)*Re[I^n]-4); Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Apr 30 2013 *) LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, -1}, {-1, 1, 1, 3, 0, 5, 3, 7}, 80] (* Harvey P. Dale, Jul 14 2019 *)
PROG
(PARI) x='x+O('x^50); Vec((-1+x+x^2+3*x^3+3*x^5+x^6+x^7+2*x^4)/((-1+x)^2*(1+x)^2*(x^2+1)^2)) \\ G. C. Greubel, Sep 20 2018 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-1+x+x^2+3*x^3+3*x^5+x^6+x^7+2*x^4)/((-1+x)^2*(1+x)^2*(x^2+1)^2))); // G. C. Greubel, Sep 20 2018
a(n) = n^2 / gcd(n+2, 4).
+10
1
0, 1, 1, 9, 8, 25, 9, 49, 32, 81, 25, 121, 72, 169, 49, 225, 128, 289, 81, 361, 200, 441, 121, 529, 288, 625, 169, 729, 392, 841, 225, 961, 512, 1089, 289, 1225, 648, 1369, 361, 1521, 800, 1681, 441, 1849, 968, 2025, 529, 2209, 1152, 2401, 625, 2601, 1352
COMMENTS
A061038(n), which appears in 4*a(n) formula, is a permutation of n^2.
Origin. In December 2010, I wrote in my 192-page Exercise Book no. 5, page 41, the array (difference table of the first row):
1 0, 1/3, 1, 9/5, 8/3, 25/7, 9/2, 49/9, ...
-1, 1/3, 2/3, 4/5, 13/15, 19/21, 13/14, 17/18, 43/45, ...
Numerators are listed in A176126, denominators are in A064038, and denominator - numerator = 2, 2, 1, 1,... ( A014695). 4/3, 1/3, 2/15, 1/15, 4/105, 1/42, 1/63, 1/90, 4/495, ...
-1, -1/5, -1/15, -1/35, -1/70, -1/126, -1/210, -1/330, -1/495, ...
where the denominators of the second row are listed in A000332. Also for those of the inverse binomial transform
1, -1, 4/3, -1, 4/5, -2/3, 4/7, -1/2, 4/9, -2/5, 4/11, -1/3, ... ?
a(n) is the (n+1)-th term of the numerators of the first row.
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).
FORMULA
a(n) = n^2/(period 4: repeat 2, 1, 4, 1).
a(4n) = 8*n^2, a(2n+1) = a(4n+2) = (2*n+1)^2.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12), n>11.
4*a(n) = (period 4: repeat 2, 1, 4, 1) * A061038(n). G.f.: -x*(x^10+x^9+9*x^8+8*x^7+22*x^6+6*x^5+22*x^4+8*x^3+9*x^2+x+1) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, May 14 2015 a(n) = ( 1 - (1/16)*(1+(-1)^n)*(5-(-1)^(n/2)) )*n^2. - Bruno Berselli, May 14 2015
EXAMPLE
a(0) = 0/2, a(1) = 1/1, a(2) = 4/4, a(3) = 9/1.
MAPLE
seq(seq((4*i+j-1)^2/[2, 1, 4, 1][j], j=1..4), i=0..30); # Robert Israel, May 14 2015
MATHEMATICA
f[n_] := Switch[ Mod[n, 4], 0, n^2/2, 1, n^2, 2, n^2/4, 3, n^2]; Array[f, 50, 0] (* or *) Table[(4 i + j - 1)^2/{2, 1, 4, 1}[[j]], {i, 0, 12}, {j, 4}] // Flatten (* after Robert Israel *) (* or *) LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 1, 9, 8, 25, 9, 49, 32, 81, 25, 121}, 53] (* or *) CoefficientList[ Series[-((x (1 + x (1 + x (9 + x (8 + x (22 + x (6 + x (22 + x (8 + x (9 + x + x^2))))))))))/(-1 + x^4)^3), {x, 0, 52}], x] (* Robert G. Wilson v, May 19 2015 *)
PROG
(PARI) concat(0, Vec(-x*(x^10 + x^9 + 9*x^8 + 8*x^7 + 22*x^6 + 6*x^5 + 22*x^4 + 8*x^3 + 9*x^2 + x + 1) / ((x-1)^3*(x+1)^3*(x^2+1)^3) + O(x^100))) \\ Colin Barker, May 14 2015 (Magma) [(1-(1/16)*(1+(-1)^n)*(5-(-1)^(n div 2)) )*n^2: n in [0..60]]; // Vincenzo Librandi, Jun 12 2015
CROSSREFS
Cf. A000290, A000332, A016754, A026741, A060819, A061038, A109008, A139098, A168077, A176895, A181900.
EXTENSIONS
Missing term (1521) inserted in the sequence by Colin Barker, May 14 2015
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