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%I #31 Mar 28 2024 10:53:29
%S 1,2,6,12,26,50,97,180,332,600,1076,1908,3361,5878,10226,17700,30510,
%T 52390,89665,153000,260376,442032,748776,1265832,2136001,3598250,
%U 6052062,10164540,17048642
%N Third column of triangle A054453.
%H G. C. Greubel, <a href="/A054454/b054454.txt">Table of n, a(n) for n = 0..1000</a>
%H Charles H. Conley and Valentin Ovsienko, <a href="https://arxiv.org/abs/2209.10426">Shadows of rationals and irrationals: supersymmetric continued fractions and the super modular group</a>, arXiv:2209.10426 [math-ph], 2022.
%H Kálmán Liptai, László Németh, Tamás Szakács, and László Szalay, <a href="https://arxiv.org/abs/2403.15053">On certain Fibonacci representations</a>, arXiv:2403.15053 [math.NT], 2024. See p. 8.
%H Gregg Musiker, Nick Ovenhouse, and Sylvester W. Zhang, <a href="https://www-users.cse.umn.edu/~swzhang/files/MOZ-FPSAC23.pdf">Double Dimers and Super Ptolemy Relations</a>, Séminaire Lotharingien de Combinatoire XX, Proc. 35th Conf. Formal Power, Series and Algebraic Combinatorics (Davis) 2023, Art. #YY. See p. 12.
%H László Németh, <a href="https://arxiv.org/abs/2403.12159">Walks on tiled boards</a>, arXiv:2403.12159 [math.CO], 2024. See p. 3.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-4,-2,2,1).
%F a(n) = A054453(n+2, 2).
%F a(2*k) = 1 + (8*n*Fibonacci(2*n+1) + 3*(2*n+1)*Fibonacci(2*n))/5.
%F a(2*k+1) = 2*(2*(2*n+1)*Fibonacci(2*(n+1)) + 3*(n+1)*Fibonacci(2*n+1))/5.
%F G.f.: ((Fib(x))^2)/(1-x^2), with Fib(x)=1/(1-x-x^2) = g.f. A000045(n+1)(Fibonacci numbers without F(0)).
%F a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3) - 2*a(n-4) + 2*a(n-5) + a(n-6) where a(0)=1, a(1)=2, a(2)=6, a(3)=12, a(4)=26, a(5)=50. - _Harvey P. Dale_, May 06 2012
%t CoefficientList[Series[(1/(1-x-x^2))^2/(1-x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,-4,-2,2,1},{1,2,6,12,26,50},30] (* _Harvey P. Dale_, May 06 2012 *)
%o (PARI) my(x='x+O('x^30)); Vec(1/((1-x^2)*(1-x-x^2)^2)) \\ _G. C. Greubel_, Jan 31 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/((1-x^2)*(1-x-x^2)^2) )); // _G. C. Greubel_, Jan 31 2019
%o (Sage) (1/((1-x^2)*(1-x-x^2)^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 31 2019
%o (GAP) a:=[1,2,6,12,26,50];; for n in [7..30] do a[n]:=2*a[n-1]+2*a[n-2] -4*a[n-3]-2*a[n-4]+2*a[n-5]+a[n-6]; od; a; # _G. C. Greubel_, Jan 31 2019
%Y Cf. A054453, A000045.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_, Apr 27 2000
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