A254962 Indices of hexagonal numbers (A000384) that are also centered pentagonal numbers (A005891).

1, 2, 12, 31, 211, 552, 3782, 9901, 67861, 177662, 1217712, 3188011, 21850951, 57206532, 392099402, 1026529561, 7035938281, 18420325562, 126254789652, 330539330551, 2265550275451, 5931287624352, 40653650168462, 106432637907781, 729500152756861

Offset: 1

Author

Colin Barker, Feb 11 2015

Keywords

• nonn
• easy

Comments

Also positive integers x in the solutions to 4*x^2 - 5*y^2 - 2*x + 5*y - 2 = 0, the corresponding values of y being A254627.

Examples

			12 is in the sequence because the 12th hexagonal number is 276, which is also the 11th 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. A000032 (Lucas numbers), A000384, A005891, A254627, A254628.

Programs

PARI

Vec(-x*(x^4+x^3-8*x^2+x+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^4+x^3-8*x^2+x+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)/8 where r = sqrt(5). - Colin Barker, Nov 25 2016

a(n+2) - a(n) = A000032(3*n + 2) if n is odd, A000032(3*n + 1) if n is even. - Diego Rattaggi, May 11 2020