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%I #36 Aug 09 2024 11:49:36
%S 1,1,3,6,12,23,43,79,143,256,454,799,1397,2429,4203,7242,12432,21271,
%T 36287,61739,104791,177476,299978,506111,852457,1433593,2407443,
%U 4037454,6762708,11314391,18909139,31569799,52657247,87751624
%N Number of certain stackings of n+1 squares on a double staircase.
%C a(n)= G_{n+1} of Turban reference eq.(3.9).
%C Equals A046854 * [1,2,3,...]. - _Gary W. Adamson_, Dec 23 2008
%C (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
%C Column 1 of A194540. - _R. H. Hardin_, Aug 28 2011
%D L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595.
%H Vincenzo Librandi, <a href="/A055244/b055244.txt">Table of n, a(n) for n = 0..1000</a>
%H Arnold Adelberg and Tamás Lengyel, <a href="https://www.researchgate.net/profile/Arnold-Adelberg/publication/382731171_NEW_RESULTS_ON_THE_2-ADIC_VALUATION_OF_THE_CENTRAL_STIRLING_NUMBERS_S2K_K/">New results on the 2-adic valuation of the central Stirling numbers S(2k, k)</a>, Harvey Mudd College (2024).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F G.f.: (1-x+x^3)/(1-x-x^2)^2. (from Turban reference eq.(3.3) with t=1).
%F 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)).
%F a(n) = A001629(n+1) + F(n-1). - _Gary W. Adamson_, Jul 27 2007
%F 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
%p 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
%t 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 *)
%t CoefficientList[Series[(1 - x + x^3)/(1 - x - x^2)^2, {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 13 2014 *)
%t LinearRecurrence[{2,1,-2,-1},{1,1,3,6},60] (* _Harvey P. Dale_, Jul 13 2022 *)
%Y Cf. A000045, A001629, A046854, A055245.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, May 10 2000