A254962 Indices of hexagonal numbers (A000384) that are also centered pentagonal numbers (A005891).

1, 2, 12, 31, 211, 552, 3782, 9901, 67861, 177662, 1217712, 3188011, 21850951, 57206532, 392099402, 1026529561, 7035938281, 18420325562, 126254789652, 330539330551, 2265550275451, 5931287624352, 40653650168462, 106432637907781, 729500152756861

Offset

1,2

Comments

Also positive integers x in the solutions to 4*x^2 - 5*y^2 - 2*x + 5*y - 2 = 0, the corresponding values of y being A254627.

Links

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

Index entries for linear recurrences with constant coefficients, signature (1,18,-18,-1,1).

Formula

a(n) = a(n-1)+18*a(n-2)-18*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+x^3-8*x^2+x+1) / ((x-1)*(x^2-4*x-1)*(x^2+4*x-1)).

a(n) = (2 + (2-r)^n - (-2-r)^n*(-2+r) + 2*(-2+r)^n + r*(-2+r)^n + (2+r)^n)/8 where r = sqrt(5). - Colin Barker, Nov 25 2016

a(n+2) - a(n) = A000032(3*n + 2) if n is odd, A000032(3*n + 1) if n is even. - Diego Rattaggi, May 11 2020

Example

12 is in the sequence because the 12th hexagonal number is 276, which is also the 11th centered pentagonal number.

Prog

(PARI) Vec(-x*(x^4+x^3-8*x^2+x+1)/((x-1)*(x^2-4*x-1)*(x^2+4*x-1)) + O(x^100))

Crossrefs

Cf. A000032 (Lucas numbers), A000384, A005891, A254627, A254628.

Sequence in context: A101177 A117625 A297763 * A139323 A225525 A240395

Adjacent sequences: A254959 A254960 A254961 * A254963 A254964 A254965

Keyword

nonn,easy

Author

Colin Barker, Feb 11 2015

Status

approved