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

Author

Wolfdieter Lang, May 10 2000

Keywords

• nonn
• easy

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

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

Crossrefs

Cf. A000045, A055244.

Programs

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 *)

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