A055245 Numerator sequence of mean length of certain stackings of n+1 squares on a double staircase.

1, 1, 5, 12, 28, 61, 127, 257, 507, 982, 1872, 3523, 6557, 12089, 22105, 40128, 72380, 129809, 231611, 411337, 727455, 1281578, 2249856, 3936935, 6868537, 11950033, 20737613, 35901300, 62014396, 106897669, 183905143, 315806321, 541372131

Offset

0,3

Comments

Denominator sequence is A055244(n).

References

L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595.

Links

Table of n, a(n) for n=0..32.

Index entries for linear recurrences with constant coefficients, signature (3, 0, -5, 0, 3, 1).

Formula

G.f.: (1-2*x+2*x^2+2*x^3-3*x^4-x^5)/(1-x-x^2)^3. (from Turban reference eq.(3.11)).

a(n) = ((5*n^2+3*n-27)*F(n)+(19*n+25)*F(n+1))/25 with F(n)=A000045(n) (Fibonacci numbers) (from Turban reference eq.(3.12)).

a(0)=1, a(1)=1, a(2)=5, a(3)=12, a(4)=28, a(5)=61, a(n)=3*a(n-1)- 5*a(n-3)+ 3*a(n-5)+a(n-6). - Harvey P. Dale, Aug 24 2014

Maple

a:= n-> (Matrix([[1, -1, 0, 2, -9, 25]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [3, 0, -5, 0, 3, 1][i] else 0 fi)^(n))[1, 1]: seq(a(n), n=0..32); # Alois P. Heinz, Aug 05 2008

Mathematica

CoefficientList[Series[(1-2x+2x^2+2x^3-3x^4-x^5)/(1-x-x^2)^3, {x, 0, 50}], x] (* or *) LinearRecurrence[{3, 0, -5, 0, 3, 1}, {1, 1, 5, 12, 28, 61}, 50] (* Harvey P. Dale, Aug 24 2014 *)

Crossrefs

Cf. A000045, A055244.

Sequence in context: A160807 A038376 A002767 * A196410 A000465 A283506

Adjacent sequences: A055242 A055243 A055244 * A055246 A055247 A055248

Keyword

nonn,easy

Author

Wolfdieter Lang, May 10 2000

Status

approved