A254782 Indices of centered hexagonal numbers (A003215) which are also centered pentagonal numbers (A005891).

1, 11, 231, 5061, 111101, 2439151, 53550211, 1175665481, 25811090361, 566668322451, 12440892003551, 273132955755661, 5996484134620981, 131649518005905911, 2890292911995309051, 63454794545890893201, 1393115187097604341361, 30585079321601404616731

Offset: 1

Author

Colin Barker, Feb 07 2015

Keywords

• nonn
• easy

Comments

Also positive integers y in the solutions to 5*x^2 - 6*y^2 - 5*x + 6*y = 0, the corresponding values of x being A133285.

The numbers (as opposed to the indices) are A133141.

Examples

			11 is in the sequence because the 11th centered hexagonal number is 331, which is also the 12th centered pentagonal number.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..746
• Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
• Index entries for linear recurrences with constant coefficients, signature (23,-23,1).

Crossrefs

Cf. A003215, A005891, A133141, A133285.

Programs

Mathematica

LinearRecurrence[{23,-23,1},{1,11,231},20] (* Harvey P. Dale, Mar 01 2022 *)

PARI

Vec(-x*(x^2-12*x+1)/((x-1)*(x^2-22*x+1)) + O(x^100))

Formula

a(n) = 23*a(n-1)-23*a(n-2)+a(n-3).

G.f.: -x*(x^2-12*x+1) / ((x-1)*(x^2-22*x+1)).

a(n) = 1/2+1/24*(11+2*sqrt(30))^(-n)*(6+sqrt(30)-(-6+sqrt(30))*(11+2*sqrt(30))^(2*n)). - Colin Barker, Mar 03 2016