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A253621 Indices of centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891). 3
1, 6, 66, 781, 9301, 110826, 1320606, 15736441, 187516681, 2234463726, 26626048026, 317278112581, 3780711302941, 45051257522706, 536834378969526, 6396961290111601, 76226701102369681, 908323451938324566, 10823654722157525106, 128975533213951976701 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 8 18:40 EDT 2022. Contains 356016 sequences. (Running on oeis4.)