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

Author

Colin Barker, Feb 03 2015

Keywords

• nonn
• easy

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.

Examples

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

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).

Crossrefs

Cf. A000217, A005891, A254627, A254628.

Programs

Mathematica

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

PARI

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

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