A254626 Indices of triangular numbers (A000217) that are also centered pentagonal numbers (A005891).

1, 3, 23, 61, 421, 1103, 7563, 19801, 135721, 355323, 2435423, 6376021, 43701901, 114413063, 784198803, 2053059121, 14071876561, 36840651123, 252509579303, 661078661101, 4531100550901, 11862575248703, 81307300336923, 212865275815561, 1459000305513721

Offset

1,2

Comments

Also positive integers x in the solutions to x^2 - 5*y^2 + x + 5*y - 2 = 0, the corresponding values of y being A254627.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,18,-18,-1,1).

Formula

a(n) = a(n-1) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5).

G.f.: -x*(x+1)^2*(x^2+1) / ((x-1)*(x^2-4*x-1)*(x^2+4*x-1)).

a(n) = (-2 + (2-r)^n - (-2-r)^n*(-2+r) + 2*(-2+r)^n + r*(-2+r)^n + (2+r)^n)/4 where r = sqrt(5). - Colin Barker, Nov 25 2016

Example

3 is in the sequence because the 3rd triangular number is 6, which is also the 2nd centered pentagonal number.

Mathematica

LinearRecurrence[{1, 18, -18, -1, 1}, {1, 3, 23, 61, 421}, 30] (* Harvey P. Dale, Jun 15 2024 *)

Prog

(PARI) Vec(-x*(x+1)^2*(x^2+1)/((x-1)*(x^2-4*x-1)*(x^2+4*x-1)) + O(x^100))

Crossrefs

Cf. A000217, A005891, A254627, A254628.

Sequence in context: A141047 A196538 A216418 * A362540 A262769 A298393

Adjacent sequences: A254623 A254624 A254625 * A254627 A254628 A254629

Keyword

nonn,easy

Author

Colin Barker, Feb 03 2015

Status

approved