OFFSET
0,3
COMMENTS
This is the second binomial transform of the octagonal numbers A000567 and the binomial transform of A084857. - Paul Barry, Jun 09 2003
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wrath of Math, 2025 is a Strange Number, YouTube video, 2025.
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).
FORMULA
a(n) = n^2 * 3^(n-1).
E.g.f.: exp(3*x)*(x+3*x^2). - Paul Barry, Jul 23 2003
From Colin Barker, Aug 10 2012: (Start)
a(n) = 9*a(n-1)-27*a(n-2)+27*a(n-3).
G.f.: x*(1+3*x)/(1-3*x)^3. (End)
G.f.: 1 + 12*x/(G(0) - 12*x), where G(k) = 1 + 12*x + 2*k*(6*x+1) + (1+3*x)*k^2 - 3*x*(k+1)^2*(k+3)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 05 2013
MATHEMATICA
a[n_] := n^2*3^(n - 1); Table[ a[n], {n, 1, 24}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric Weinhandl (eweinhandl(AT)msn.com), May 01 2002
EXTENSIONS
Edited and extended by Robert G. Wilson v, May 04 2002
a(0)=0 prepended by Alois P. Heinz, Jan 03 2025
STATUS
approved