login
A254962
Indices of hexagonal numbers (A000384) that are also centered pentagonal numbers (A005891).
3
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.
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
KEYWORD
nonn,easy
AUTHOR
Colin Barker, Feb 11 2015
STATUS
approved