A143166 -id:A143166 - cp's OEIS frontend

A118465 a(n) = 8*n^3 + n.

0, 9, 66, 219, 516, 1005, 1734, 2751, 4104, 5841, 8010, 10659, 13836, 17589, 21966, 27015, 32784, 39321, 46674, 54891, 64020, 74109, 85206, 97359, 110616, 125025, 140634, 157491, 175644, 195141, 216030, 238359, 262176, 287529, 314466, 343035, 373284, 405261

Offset: 0

Mohamed Bouhamida, May 16 2006, Oct 02 2007

(8*n^3 + n, 8*n^3 - n) solves the Diophantine equation 2*(X-Y)^3-(X+Y)=0.
(m*(2n)^k+n, m*(2n)^k-n) solves the Diophantine equation: 2m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 and where m is a positive integer. Also ((m*n^k+n)/2, (m*n^k-n)/2) solves the Diophantine equation: m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 where m is an odd number.
24*a(n) = (4*n+1)^3 + (4*n)^3 + (4*n-1)^3. - Bruno Berselli, May 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006003, A081585, A143166.

[8*n^3 + n: n in [0..30]]; // Wesley Ivan Hurt, May 13 2014

A118465:=n->8*n^3 + n; seq(A118465(n), n=0..30); # Wesley Ivan Hurt, May 13 2014

Table[8 n^3 + n, {n, 0, 35}]
CoefficientList[Series[3 x (x + 3) (3 x + 1)/(-1 + x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,9,66,219},40] (* Harvey P. Dale, Feb 01 2023 *)

G.f.: 3*x*(x+3)*(3*x+1)/(-1+x)^4. - R. J. Mathar, Nov 14 2007
a(n) = n*A081585(n). - Vincenzo Librandi, May 13 2014
From Elmo R. Oliveira, Aug 07 2025: (Start)
E.g.f.: exp(x)*x*(9 + 24*x + 8*x^2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*A143166(n). (End)

Edited by Stefan Steinerberger, Jul 24 2007

A329020 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( Sum_{j=1..k} x_j^(2j-1) + x_j^(-(2j-1)) )^(2*n).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 44, 20, 0, 1, 8, 146, 580, 70, 0, 1, 10, 344, 4332, 8092, 252, 0, 1, 12, 670, 18152, 135954, 116304, 924, 0, 1, 14, 1156, 55252, 1012664, 4395456, 1703636, 3432, 0, 1, 16, 1834, 137292, 4816030, 58199208, 144840476, 25288120, 12870, 0

Offset: 0

Seiichi Manyama, Nov 02 2019

			(x^3 + x + 1/x + 1/x^3)^2 = x^6 + 2*x^4 + 3*x^2 + 4 + 3/x^2 + 2/x^4 + 1/x^6. So T(1,2) = 4.
Square array begins:
   1,   1,      1,       1,        1,         1, ...
   0,   2,      4,       6,        8,        10, ...
   0,   6,     44,     146,      344,       670, ...
   0,  20,    580,    4332,    18152,     55252, ...
   0,  70,   8092,  135954,  1012664,   4816030, ...
   0, 252, 116304, 4395456, 58199208, 432457640, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Columns k=0-3 give A000007, A000984, A005721, A063419.
Rows n=0-2 give A000012, A005843, 2*A143166.
Main diagonal gives A329021.
Cf. A077042.

T[n_, 0] = Boole[n == 0]; T[n_, k_] := Sum[(-1)^j * Binomial[2*n, j] * Binomial[(2*k + 1)*n - 2*k*j - 1, (2*k - 1)*n - 2*k*j], {j, 0, Floor[(2*k - 1)*n/(2*k)]}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

T(n,k) = Sum_{j=0..floor((2*k-1)*n/(2*k))} (-1)^j * binomial(2*n,j) * binomial((2*k+1)*n-2*k*j-1,(2*k-1)*n-2*k*j) for k > 0.

cp's OEIS Frontend

A118465 a(n) = 8*n^3 + n.

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A329020 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( Sum_{j=1..k} x_j^(2j-1) + x_j^(-(2j-1)) )^(2*n).

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cp's OEIS Frontend

A118465 a(n) = 8*n^3 + n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A329020 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( Sum_{j=1..k} x_j^(2*j-1) + x_j^(-(2*j-1)) )^(2*n).

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Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

Formula

A329020 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( Sum_{j=1..k} x_j^(2j-1) + x_j^(-(2j-1)) )^(2*n).