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

1,2

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.

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).

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A003215, A069099, A253457, A253546.

Sequence in context: A166743 A212733 A183803 * A012492 A296395 A183440

Adjacent sequences: A253455 A253456 A253457 * A253459 A253460 A253461

Keyword

nonn,easy

Author

Colin Barker, Jan 01 2015

Status

approved