A253621 Indices of centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).

1, 6, 66, 781, 9301, 110826, 1320606, 15736441, 187516681, 2234463726, 26626048026, 317278112581, 3780711302941, 45051257522706, 536834378969526, 6396961290111601, 76226701102369681, 908323451938324566, 10823654722157525106, 128975533213951976701

Offset: 1

Author

Colin Barker, Jan 06 2015

Keywords

• nonn
• easy

Comments

Also positive integers y in the solutions to 5*x^2 - 7*y^2 - 5*x + 7*y = 0, the corresponding values of x being A133272.

Examples

			6 is in the sequence because the 6th centered heptagonal number is 106, which is also the 7th centered pentagonal number.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..930
• 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 (13,-13,1).

Crossrefs

Cf. A005891, A069099, A133272, A253622.

Programs

Magma

I:=[1,6]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2)-5: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016

Mathematica

RecurrenceTable[{a[1] == 1, a[2] == 6, a[n] == 12 a[n-1] - a[n-2] - 5}, a, {n, 20}] (* Vincenzo Librandi, Mar 05 2016 *)

PARI

Vec(-x*(x^2-7*x+1)/((x-1)*(x^2-12*x+1)) + O(x^100))

Formula

a(n) = 13*a(n-1)-13*a(n-2)+a(n-3).

G.f.: -x*(x^2-7*x+1) / ((x-1)*(x^2-12*x+1)).

a(n) = (14-(-7+sqrt(35))*(6+sqrt(35))^n+(6-sqrt(35))^n*(7+sqrt(35)))/28. - Colin Barker, Mar 05 2016

a(n) = 12*a(n-1) - a(n-2) - 5. - Vincenzo Librandi, Mar 05 2016

a(n) = (5*a(n-1) + a(n-1)^2) / a(n-2), n >= 3. - Seiichi Manyama, Aug 11 2016