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

1,2

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.

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).

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A005891, A069099, A133272, A253622.

Sequence in context: A129554 A165229 A247740 * A127857 A127858 A173535

Adjacent sequences: A253618 A253619 A253620 * A253622 A253623 A253624

Keyword

nonn,easy

Author

Colin Barker, Jan 06 2015

Status

approved