OFFSET
0,3
COMMENTS
a(n)= G_{n+1} of Turban reference eq.(3.9).
Equals A046854 * [1,2,3,...]. - Gary W. Adamson, Dec 23 2008
(1 + x + 3x^2 + 6x^3 + ...) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + ...) * (1 + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + ...). -Gary W. Adamson, Jul 27 2010
Column 1 of A194540. - R. H. Hardin, Aug 28 2011
REFERENCES
L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Arnold Adelberg and Tamás Lengyel, New results on the 2-adic valuation of the central Stirling numbers S(2k, k), Harvey Mudd College (2024).
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).
FORMULA
G.f.: (1-x+x^3)/(1-x-x^2)^2. (from Turban reference eq.(3.3) with t=1).
a(n) = ((n+5)*F(n+1)+(2*n-3)*F(n))/5 with F(n)=A000045(n) (Fibonacci numbers) (from Turban reference eq.(3.9)).
a(n) = A001629(n+1) + F(n-1). - Gary W. Adamson, Jul 27 2007
a(n) = (((n-4)*n-6)*a(n-2) + ((n-5)*n-11)*a(n-1)) / ((n-6)*n-1). - Jean-François Alcover, Mar 11 2014
MAPLE
a:= n-> (Matrix([[1, -1, 2, -4]]). Matrix(4, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -2, -1][i] else 0 fi)^(n))[1, 1] ; seq (a(n), n=0..33); # Alois P. Heinz, Aug 05 2008
MATHEMATICA
a[0] = a[1] = 1; a[n_] := a[n] = (((n-4)*n-6)*a[n-2] + ((n-5)*n-11)*a[n-1]) / ((n-6)*n-1); Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 11 2014 *)
CoefficientList[Series[(1 - x + x^3)/(1 - x - x^2)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 13 2014 *)
LinearRecurrence[{2, 1, -2, -1}, {1, 1, 3, 6}, 60] (* Harvey P. Dale, Jul 13 2022 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, May 10 2000
STATUS
approved