OFFSET
0,2
COMMENTS
Binomial transform of (1, 8, 9, 0, 0, 0, ...).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjić, Two Enumerative Functions
Richard J. Mathar, Bivariate generating functions enumerating non-bonding dominoes on rectangular boards, arXiv:2404.18806 (2024) Table 2.
Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 7.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = C(n, 0) + 8*C(n, 1) + 9*C(n, 2).
a(n) = (9*n^2 + 7*n + 2)/2.
G.f.: (1 + 6*x + 2*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), for n > 2. a(n) = right term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 0 0 / 3 1 0 / 5 3 1]. M^n * [1 1 1] = [1 3n+1 a(n)]. - Gary W. Adamson, Dec 22 2004
a(n) = 9*n + a(n-1) - 1 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
E.g.f.: exp(x)*(2 + 16*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {1, 9, 26}, 50] (* Harvey P. Dale, Aug 13 2014 *)
CoefficientList[Series[(1 + 6 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 14 2014 *)
PROG
(Magma) [(9*n^2 + 7*n + 2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 14 2014
(PARI) a(n)=(9*n^2+7*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 15 2003
STATUS
approved