A253458 Indices of centered heptagonal numbers (A069099) which are also centered hexagonal numbers (A003215).

1, 13, 325, 8425, 218713, 5678101, 147411901, 3827031313, 99355402225, 2579413426525, 66965393687413, 1738520822446201, 45134575989913801, 1171760454915312613, 30420637251808214125, 789764808092098254625, 20503464373142746406113, 532300308893619308304301

Offset: 1

Author

Colin Barker, Jan 01 2015

Keywords

• nonn
• easy

Comments

Also positive integers y in the solutions to 6*x^2 - 7*y^2 - 6*x + 7*y = 0, the corresponding values of x being A253457.

Examples

			13 is in the sequence because the 13th centered heptagonal number is 547, which is also the 14th centered hexagonal number.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..708
• 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 (27,-27,1).

Crossrefs

Cf. A003215, A069099, A253457, A253546.

Programs

Mathematica

LinearRecurrence[{27,-27,1},{1,13,325},20] (* Harvey P. Dale, Oct 13 2022 *)

PARI

Vec(-x*(x^2-14*x+1)/((x-1)*(x^2-26*x+1)) + O(x^100))

Formula

a(n) = 27*a(n-1)-27*a(n-2)+a(n-3).

G.f.: -x*(x^2-14*x+1) / ((x-1)*(x^2-26*x+1)).

a(n) = 1/2+(13+2*sqrt(42))^(-n)*(7+sqrt(42)-(-7+sqrt(42))*(13+2*sqrt(42))^(2*n))/28. - Colin Barker, Mar 03 2016