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A055244 Number of certain stackings of n+1 squares on a double staircase. 8
1, 1, 3, 6, 12, 23, 43, 79, 143, 256, 454, 799, 1397, 2429, 4203, 7242, 12432, 21271, 36287, 61739, 104791, 177476, 299978, 506111, 852457, 1433593, 2407443, 4037454, 6762708, 11314391, 18909139, 31569799, 52657247, 87751624 (list; graph; refs; listen; history; text; internal format)
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

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 *)

CROSSREFS

Cf. A000045, A001629, A046854, A055245.

Sequence in context: A181844 A162506 A227681 * A089068 A018180 A079735

Adjacent sequences:  A055241 A055242 A055243 * A055245 A055246 A055247

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, May 10 2000

STATUS

approved

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Last modified November 14 09:49 EST 2018. Contains 317182 sequences. (Running on oeis4.)