A254332 Indices of centered pentagonal numbers (A005891) which are also squares (A000290).

1, 3, 22, 96, 817, 3627, 31006, 137712, 1177393, 5229411, 44709910, 198579888, 1697799169, 7540806315, 64471658494, 286352060064, 2448225223585, 10873837476099, 92968086837718, 412919472031680, 3530339074609681, 15680066099727723, 134059916748330142

Offset: 1

Author

Colin Barker, Jan 28 2015

Keywords

• nonn
• easy

Comments

Also positive integers y in the solutions to 2*x^2 - 5*y^2 + 5*y - 2 = 0, the corresponding values of x being A129557.

Examples

			3 is in the sequence because the 3rd centered pentagonal number is 16, which is also the 4th square number.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,38,-38,-1,1).

Crossrefs

Cf. A000290, A005891, A129557, A254333.

Programs

Mathematica

LinearRecurrence[{1,38,-38,-1,1},{1,3,22,96,817},30] (* Harvey P. Dale, Mar 27 2017 *)

PARI

Vec(x*(2*x^3+19*x^2-2*x-1) / ((x-1)*(x^2-6*x-1)*(x^2+6*x-1)) + O(x^100))

Formula

a(n) = a(n-1) + 38*a(n-2) - 38*a(n-3) - a(n-4) + a(n-5).

G.f.: x*(2*x^3 + 19*x^2 - 2*x - 1) / ((x-1)*(x^2 - 6*x - 1)*(x^2 + 6*x - 1)).

a(n) = (1/40)*(20 - b^n*(19 + 3*b) + (3 + b)*c^n - (b^n*(3 + b) + (1 - 3*b)*c^n)*(-1)^n) with b = sqrt(10) - 3 and c = sqrt(10) + 3. - Alan Michael Gómez Calderón, Jul 02 2024