A253175 Indices of hexagonal numbers (A000384) which are also centered hexagonal numbers (A003215).

1, 7, 67, 661, 6541, 64747, 640927, 6344521, 62804281, 621698287, 6154178587, 60920087581, 603046697221, 5969546884627, 59092422149047, 584954674605841, 5790454323909361, 57319588564487767, 567405431320968307, 5616734724645195301, 55599941815130984701

Offset: 1

Author

Colin Barker, Jan 08 2015

Keywords

• nonn
• easy

Comments

Also positive integers x in the solutions to 4*x^2-6*y^2-2*x+6*y-2 = 0, the corresponding values of y being A253475.

Examples

			7 is in the sequence because the 7th hexagonal number is 91, which is also the 6th centered hexagonal number.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Crossrefs

Cf. A000384, A003215, A253475, A006244, A350923.

Programs

Mathematica

LinearRecurrence[{11, -11, 1}, {1, 7, 67}, 25] (* Paolo Xausa, May 30 2025 *)

PARI

Vec(-x*(x^2-4*x+1)/((x-1)*(x^2-10*x+1)) + O(x^100))

Formula

a(n) = 11*a(n-1)-11*a(n-2)+a(n-3).

G.f.: -x*(x^2-4*x+1) / ((x-1)*(x^2-10*x+1)).

a(n) = (2+(5-2*sqrt(6))^n*(3+sqrt(6))-(-3+sqrt(6))*(5+2*sqrt(6))^n)/8. - Colin Barker, Mar 05 2016

4*a(n) = 1+3*A072256(n). - R. J. Mathar, Feb 07 2022

a(n) = A350923(n)/2. - Paolo Xausa, May 30 2025