A253654 Indices of pentagonal numbers (A000326) which are also centered pentagonal numbers (A005891).

1, 6, 46, 361, 2841, 22366, 176086, 1386321, 10914481, 85929526, 676521726, 5326244281, 41933432521, 330141215886, 2599196294566, 20463429140641, 161108236830561, 1268402465503846, 9986111487200206, 78620489432097801, 618977803969582201, 4873201942324559806

Offset

1,2

Comments

Also positive integers x in the solutions to 3*x^2-5*y^2-x+5*y-2 = 0, the corresponding values of y being A253470.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Formula

a(n) = 9*a(n-1)-9*a(n-2)+a(n-3).

G.f.: -x*(x^2-3*x+1) / ((x-1)*(x^2-8*x+1)).

a(n) = (2-(-5+sqrt(15))*(4+sqrt(15))^n+(4-sqrt(15))^n*(5+sqrt(15)))/12. - Colin Barker, Mar 03 2016

Example

6 is in the sequence because the 6th pentagonal number is 51, which is also the 5th centered pentagonal number.

Mathematica

LinearRecurrence[{9, -9, 1}, {1, 6, 46}, 30] (* Harvey P. Dale, Nov 12 2017 *)

Prog

(PARI) Vec(-x*(x^2-3*x+1)/((x-1)*(x^2-8*x+1)) + O(x^100))

Crossrefs

Cf. A000326, A005891, A128917, A253470.

Sequence in context: A155598 A190005 A334609 * A301421 A288689 A271933

Adjacent sequences: A253651 A253652 A253653 * A253655 A253656 A253657

Keyword

nonn,easy

Author

Colin Barker, Jan 07 2015

Status

approved