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

%I A055245 #14 Jun 28 2023 22:27:27
%S A055245 1,1,5,12,28,61,127,257,507,982,1872,3523,6557,12089,22105,40128,
%T A055245 72380,129809,231611,411337,727455,1281578,2249856,3936935,6868537,
%U A055245 11950033,20737613,35901300,62014396,106897669,183905143,315806321,541372131
%N A055245 Numerator sequence of mean length of certain stackings of n+1 squares on a double staircase.
%C A055245 Denominator sequence is A055244(n).
%D A055245 L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595.
%H A055245 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3, 0, -5, 0, 3, 1).
%F A055245 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)).
%F A055245 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)).
%F A055245 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
%p A055245 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
%t A055245 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 *)
%Y A055245 Cf. A000045, A055244.
%K A055245 nonn,easy
%O A055245 0,3
%A A055245 _Wolfdieter Lang_, May 10 2000