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

1,2

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.

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).

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A003215, A005891, A133141, A133285.

Sequence in context: A077736 A068122 A015287 * A169960 A045757 A144773

Adjacent sequences: A254779 A254780 A254781 * A254783 A254784 A254785

Keyword

nonn,easy

Author

Colin Barker, Feb 07 2015

Status

approved