Displaying 1-10 of 18 results found.
Digital root of A069778(n-1) = n^3 - n^2 + 1, n >= 1. Repeat(1, 6, 3, 7, 6, 6, 4, 6, 9).
+20
1
1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9
COMMENTS
Periodic with cycle of 9: {1, 6, 3, 7, 6, 6, 4, 6, 9}.
The decimal expansion of 54588823/333333333 = 0.repeat(163766469).
FORMULA
a(n) = digital root of n^3 - n^2 + n.
EXAMPLE
For a(3) = 3 because 3^3 - 3^2 + 3 = 27 - 9 + 3 = 21 with digit sum 3 which is also the digital root of 21.
MATHEMATICA
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 6, 3, 7, 6, 6, 4, 6, 9}, 108] (* Ray Chandler, Jul 25 2016 *)
PROG
(PARI) DR(n)=s=sumdigits(n); while(s>9, s=sumdigits(s)); s
for(n=1, 100, print1(DR(abs(n^2-n-n^3)), ", ")) \\ Derek Orr, Dec 30 2014
EXTENSIONS
Edited: name changed; formula, comment and example rewritten; digital root link added. - Wolfdieter Lang, Jan 05 2015
Connell sequence: 1 odd, 2 even, 3 odd, ...
(Formerly M0962 N0359)
+10
38
1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122
COMMENTS
a(t_n) = a(n(n+1)/2) = n^2 relates squares to triangular numbers. - Daniel Forgues
The natural numbers not included are A118011(n) = 4n - a(n) as n=1,2,3,... - Paul D. Hanna, Apr 10 2006 As a triangle with row sums = A069778 (1, 6, 21, 52, 105, ...): /Q 1;/Q 2, 4;/Q 5, 7, 9;/Q 10, 12, 14, 16;/Q ... . - Gary W. Adamson, Sep 01 2008 The triangle sums, see A180662 for their definitions, link the Connell sequence A001614 as a triangle with six sequences, see the crossrefs. - Johannes W. Meijer, May 20 2011
REFERENCES
C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 276.
C. A. Pickover, The Mathematics of Oz, Chapter 39, Camb. Univ. Press UK 2002.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Ian Connell and Andrew Korsak, Problem E1382, Amer. Math. Monthly, 67 (1960), 380.
FORMULA
a(n) = 2*n - floor( (1+ sqrt(8*n-7))/2 ).
a(1) = 1; then a(n) = a(n-1)+1 if a(n-1) is a square, a(n) = a(n-1)+2 otherwise. For example, a(21)=36 is a square therefore a(22)=36+1=37 which is not a square so a(23)=37+2=39 ... - Benoit Cloitre, Feb 07 2007 Let the sequence be written in the form of the triangle in the EXAMPLE section below and let a(n) and a(n+1) belong to the same row of the triangle. Then a(n)*a(n+1) + 1 = a( A000217( A118011(n))) = A000290( A118011(n)). - Ivan N. Ianakiev, Aug 16 2013 G.f. 2*x/(1-x)^2 - (x/(1-x))*sum(n>=0, x^(n*(n+1)/2))
= 2*x/(1-x)^2 - (Theta2(0,x^(1/2)))*x^(7/8)/(2*(1-x)) where Theta2 is a Jacobi theta function.
a(n) = 2*n-1 - Sum(i=0..n-2, A023531(i)). (End)
EXAMPLE
Written as a triangle the sequence begins:
1;
2, 4;
5, 7, 9;
10, 12, 14, 16;
17, 19, 21, 23, 25;
26, 28, 30, 32, 34, 36;
37, 39, 41, 43, 45, 47, 49;
50, 52, 54, 56, 58, 60, 62, 64;
65, 67, 69, 71, 73, 75, 77, 79, 81;
82, 84, 86, 88, 90, 92, 94, 96, 98, 100;
...
Right border gives A000290, n >= 1. (End)
MATHEMATICA
lst={}; i=0; For[j=1, j<=4!, a=i+1; b=j; k=0; For[i=a, i<=9!, k++; AppendTo[lst, i]; If[k>=b, Break[]]; i=i+2]; j++ ]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *) row[n_] := 2*Range[n+1]+n^2-1; Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)
PROG
(Haskell)
a001614 n = a001614_list !! (n-1)
a001614_list = f 0 0 a057211_list where
f c z (x:xs) = z' : f x z' xs where z' = z + 1 + 0 ^ abs (x - c)
(Python)
from math import isqrt
def A001614(n): return (m:=n<<1)-(k:=isqrt(m))-int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022
CROSSREFS
Triangle columns: A002522, A117950 (n>=1), A117951 (n>=2), A117619 (n>=3), A154533 (n>=5), A000290 (n>=1), A008865 (n>=2), A028347 (n>=3), A028878 (n>=1), A028884 (n>=2), A054569 [T(2*n,n)]. Triangle sums (see the comments): A069778 (Row1), A190716 (Row2), A058187 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Fi1, Fi2, Ze1 and Ze2), A000292 (Related to Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4), A190717 (Related to Ca1, Ca2, Ze3, Ze4), A190718 (Related to Gi1 and Gi2). (End)
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Mar 16 2001
Array of q-factorial numbers n!_q, read by ascending antidiagonals.
+10
19
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 21, 4, 1, 1, 1, 120, 315, 52, 5, 1, 1, 1, 720, 9765, 2080, 105, 6, 1, 1, 1, 5040, 615195, 251680, 8925, 186, 7, 1, 1, 1, 40320, 78129765, 91611520, 3043425, 29016, 301, 8, 1, 1
FORMULA
T(n,q) = Product_{k=1..n} (q^k - 1) / (q - 1).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 6, 21, 52, 105, 186, 301, ...
1, 24, 315, 2080, 8925, 29016, 77959, ...
1, 120, 9765, 251680, 3043425, 22661496, 121226245, ...
...
MAPLE
A069777 := proc(n, k) local n1: mul( A104878(n1, k), n1=k..n-1) end: A104878 := proc(n, k): if k = 0 then 1 elif k=1 then n elif k>=2 then (k^(n-k+1)-1)/(k-1) fi: end: seq(seq( A069777(n, k), k=0..n), n=0..9); # Johannes W. Meijer, Aug 21 2011
nmax:=9: T(0, 0):=1: for n from 1 to nmax do T(n, 0):=1: T(n, 1):= (n-1)! od: for q from 2 to nmax do for n from 0 to nmax do T(n+q, q) := product((q^k - 1)/(q - 1), k= 1..n) od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od; seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Aug 21 2011 # alternative Maple program:
T:= proc(n, k) option remember; `if`(n<2, 1,
T(n-1, k)*`if`(k=1, n, (k^n-1)/(k-1)))
end:
MATHEMATICA
(* Returns the rectangular array *) Table[Table[QFactorial[n, q], {q, 0, 6}], {n, 0, 6}] (* Geoffrey Critzer, May 21 2017 *)
PROG
(PARI) T(n, q)=prod(k=1, n, ((q^k - 1) / (q - 1))) \\ Andrew Howroyd, Feb 19 2018
CROSSREFS
Columns q=0..11 are A000012, A000142, A005329, A015001, A015002, A015004, A015005, A015006, A015007, A015008, A015009, A015011.
Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.
+10
9
1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 9, 16, 25, 1, 5, 12, 22, 35, 51, 1, 6, 15, 28, 45, 66, 91, 1, 7, 18, 34, 55, 81, 112, 148, 1, 8, 21, 40, 65, 96, 133, 176, 225, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451
FORMULA
Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.
T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - Michel Marcus, Jun 22 2015
EXAMPLE
The array starts
1 1 3 10 ...
1 2 6 16 ...
1 3 9 22 ...
1 4 12 28 ...
The triangle starts
1;
1, 1;
1, 2, 3;
1, 3, 6, 10;
1, 4, 9, 16, 25;
...
MATHEMATICA
Table[PolygonalNumber[n, i], {n, 0, 10}, {i, n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* Harvey P. Dale, Aug 27 2016 *)
PROG
(PARI) tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", "); ); print(); ); } \\ Michel Marcus, Jun 22 2015 (Magma) [[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // G. C. Greubel, Oct 13 2018 (Sage) [[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # G. C. Greubel, Aug 14 2019 (GAP) Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # G. C. Greubel, Aug 14 2019
CROSSREFS
Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441. Antidiagonals are composed of n-gonal numbers.
a(n) = n*(5*n^2-8*n+5)/2.
+10
8
0, 1, 9, 39, 106, 225, 411, 679, 1044, 1521, 2125, 2871, 3774, 4849, 6111, 7575, 9256, 11169, 13329, 15751, 18450, 21441, 24739, 28359, 32316, 36625, 41301, 46359, 51814, 57681, 63975, 70711, 77904, 85569, 93721, 102375, 111546, 121249, 131499, 142311, 153700
COMMENTS
Sequences of the type b(m)+m*b(m-1), where b is a polygonal number:
FORMULA
G.f.: x*(1+5*x+9*x^2)/(1-x)^4.
MATHEMATICA
Table[n (5 n^2 - 8 n + 5)/2, {n, 0, 40}]
CoefficientList[Series[x (1 + 5 x + 9 x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 9, 39}, 50] (* Harvey P. Dale, May 19 2017 *)
PROG
(Magma) [n*(5*n^2-8*n+5)/2: n in [0..40]];
(Magma) I:=[0, 1, 9, 39]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013
a(n) = n^3 + 2*n^2 + 4*n + 1.
+10
8
1, 8, 25, 58, 113, 196, 313, 470, 673, 928, 1241, 1618, 2065, 2588, 3193, 3886, 4673, 5560, 6553, 7658, 8881, 10228, 11705, 13318, 15073, 16976, 19033, 21250, 23633, 26188, 28921, 31838, 34945, 38248, 41753, 45466, 49393, 53540, 57913, 62518, 67361, 72448
COMMENTS
Numbers of the type (m+1)^3 - (m-1)*m. Similar sequences are: A069778 with the closed form (m+1)^3 - m*(m+1), A152015 with (m+1)^3 - (m+1)*(m+2).
FORMULA
O.g.f.: (1 + 4*x - x^2 + 2*x^3)/(1 - x)^4.
E.g.f.: (1 + 7*x + 5*x^2 + x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
MATHEMATICA
Table[n^3 + 2 n^2 + 4 n + 1, {n, 0, 40}]
PROG
(Magma) [n^3+2*n^2+4*n+1: n in [0..50]];
(PARI) x='x+O('x^99); Vec((1+4*x-x^2+2*x^3)/(1-x)^4) \\ Altug Alkan, Apr 01 2016 (Python) for i in range(0, 100):print(i**3+2*i**2+4*i+1) # Soumil Mandal, Apr 02 2016
-1, -3, -1, 11, 39, 89, 167, 279, 431, 629, 879, 1187, 1559, 2001, 2519, 3119, 3807, 4589, 5471, 6459, 7559, 8777, 10119, 11591, 13199, 14949, 16847, 18899, 21111, 23489, 26039, 28767, 31679, 34781, 38079, 41579, 45287, 49209, 53351, 57719
Array read by antidiagonals: number of {112,221}-avoiding words.
+10
4
1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 6, 21, 16, 5, 1, 6, 33, 52, 25, 6, 1, 6, 33, 124, 105, 36, 7, 1, 6, 33, 196, 345, 186, 49, 8, 1, 6, 33, 196, 825, 786, 301, 64, 9, 1, 6, 33, 196, 1305, 2586, 1561, 456, 81, 10, 1, 6, 33, 196, 1305, 6186, 6601, 2808, 657, 100, 11
COMMENTS
A(n,k) is the number of n-long k-ary words that simultaneously avoid the patterns 112 and 221.
FORMULA
A(n, k) = k!*binomial(n, k) + Sum_{j=1..k-1} j*j!*binomial(n, j), for 2 <= k <= n, otherwise Sum_{j=1..n} j*j!*binomial(n, j), with A(1, k) = 1 and A(n, 1) = n.
T(n, k) = A(k, n-k+1).
T(n, 1) = 1.
T(n, n) = n.
T(n, n-1) = (n-1)^2.
T(2*n, n) = A093964(n), for n >= 1. (End)
EXAMPLE
Array, A(n, k), begins as:
1, 1, 1, 1, 1, 1, 1 ... 1* A000012(k); 2, 4, 6, 6, 6, 6, 6 ... 2* A158799(k-1); 3, 9, 21, 33, 33, 33, 33 ... ;
4, 16, 52, 124, 196, 196, 196 ... ;
5, 25, 105, 345, 825, 1305, 1305 ... ;
6, 36, 186, 786, 2586, 6186, 9786 ... ;
7, 49, 301, 1561, 6601, 21721, 51961 ... ;
Antidiagonal triangle, T(n, k), begins as:
1;
1, 2;
1, 4, 3;
1, 6, 9, 4;
1, 6, 21, 16, 5;
1, 6, 33, 52, 25, 6;
1, 6, 33, 124, 105, 36, 7;
1, 6, 33, 196, 345, 186, 49, 8;
1, 6, 33, 196, 825, 786, 301, 64, 9;
1, 6, 33, 196, 1305, 2586, 1561, 456, 81, 10;
MATHEMATICA
A[n_, k_]:= A[n, k]= If[n==1, 1, If[k==1, n, If[2<=k<n+1, (1-k)*k!*Binomial[n, k] + Sum[j*j!*Binomial[n, j], {j, k}], Sum[j*j!*Binomial[n, j], {j, n}] ]]];
T[n_, k_]:= A[k, n-k+1];
Table[T[k, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Dec 29 2021 *)
PROG
(PARI) A(n, k) = if(n >= k+1, sum(j=1, k, j*j!*binomial(k, j)), if(n<2, if(n<1, 0, k), n!*binomial(k, n) + sum(j=1, n-1, j*j!*binomial(k, j))));
T(n, k) = A(n-k+1, k);
for(n=1, 15, for(k=1, n, print1(T(n, k), ", ") ) )
(Sage)
@CachedFunction
def A(n, k):
if (n==1): return 1
elif (k==1): return n
elif (2 <= k < n+1): return factorial(k)*binomial(n, k) + sum( j*factorial(j)*binomial(n, j) for j in (1..k-1) )
else: return sum( j*factorial(j)*binomial(n, j) for j in (1..n) )
def T(n, k): return A(k, n-k+1)
flatten([[T(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Dec 29 2021
q-factorial numbers 4!_q.
+10
3
1, 24, 315, 2080, 8925, 29016, 77959, 182400, 384345, 746200, 1356531, 2336544, 3847285, 6097560, 9352575, 13943296, 20276529, 28845720, 40242475, 55168800, 74450061, 99048664, 130078455, 168819840, 216735625, 275487576, 346953699, 433246240, 536730405, 660043800
FORMULA
a(n) = (n + 1)*(n^2 + n + 1)*(n^3 + n^2 + n + 1).
G.f.: (1 + 17*x + 8*x^2*(21 + 43*x) + 5*x^4*(35 + 3*x))/(1 - x)^7. - Arkadiusz Wesolowski, Nov 01 2012
MATHEMATICA
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 24, 315, 2080, 8925, 29016, 77959}, 30] (* Harvey P. Dale, Aug 30 2020 *)
q-factorial numbers 5!_q.
+10
3
1, 120, 9765, 251680, 3043425, 22661496, 121226245, 510902400, 1799118945, 5507702200, 15072415941, 37630041120, 87029433985, 188664603960, 386925380325, 756298318336, 1417430759745, 2559798038520, 4472991338725, 7589075296800, 12538953723681
LINKS
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
FORMULA
a(n) = (n + 1)*(n^2 + n + 1)*(n^3 + n^2 + n + 1)*(n^4 + n^3 + n^2 + n + 1).
G.f.: (1 + x*(109 + x*(8500 + x*(150700 + x*(792550 + x*(1454134 + x*(978436 + 5*x*(45788 + x*(3053 + 33*x)))))))))/(1 - x)^11.
MATHEMATICA
Table[QFactorial[5, n], {n, 0, 20}]
Join[{1}, With[{f=Times@@Table[Total[n^Range[0, i]], {i, 4}]}, Table[f, {n, 20}]]] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 120, 9765, 251680, 3043425, 22661496, 121226245, 510902400, 1799118945, 5507702200, 15072415941}, 30] (* Harvey P. Dale, Sep 04 2017 *)
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