OFFSET
2,2
REFERENCES
J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
LINKS
G. C. Greubel, Table of n, a(n) for n = 2..500
Index entries for linear recurrences with constant coefficients, signature (3,6,-8)
FORMULA
G.f.: x^2/((1-x)*(1+2*x)*(1-4*x)).
From Vincenzo Librandi, Mar 20 2011: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) + (-1)^n *2^(n-2), n >= 4.
a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3), n >= 3. (End)
a(n) = (1/18)*(4^n - 2 + (-1)^n*2^n). - R. J. Mathar, Mar 21 2011
E.g.f.: 2*exp(x)*sinh(3*x/2)^2/9. - Stefano Spezia, Apr 25 2025
MATHEMATICA
Join[{a=1, b=3}, Table[c=2*b+8*a+1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011 *)
Table[QBinomial[n, 2, -2], {n, 2, 25}] (* G. C. Greubel, Jul 30 2016 *)
PROG
(SageMath) [gaussian_binomial(n, 2, -2) for n in range(2, 25)] # Zerinvary Lajos, May 28 2009
(PARI) a(n)=(4^n - 2 + (-1)^n*2^n)/18 \\ Charles R Greathouse IV, Jul 30 2016
(Python)
def A015249(n): return ((m:=1<<n)|1)//3*((m>>1|1)//3) # Chai Wah Wu, Apr 25 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Olivier Gérard, Dec 11 1999
STATUS
approved