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%I #19 Sep 08 2022 08:45:28
%S 1,1,1,1,3,3,11,11,41,41,153,153,571,571,2131,2131,7953,7953,29681,
%T 29681,110771,110771,413403,413403,1542841,1542841,5757961,5757961,
%U 21489003,21489003,80198051,80198051,299303201,299303201,1117014753
%N Expansion of x*(1 + x)*(1 - 3*x^2)/(1 - 4*x^2 + x^4).
%H G. C. Greubel, <a href="/A122573/b122573.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-1).
%F G.f.: x*(1 + x)*(1 - 3*x^2)/(1 - 4*x^2 + x^4).
%F a(n) = 3*b(n) + b(n-1) - 11*b(n-2) - 3*b(n-3), where a(0) = a(1) = 1, b(n) = (1/2)*(1 + (-1)^n)*c((n+2)/2), and c(n) = ((2+sqrt(3))^n - (2-sqrt(3))^n)/(2*sqrt(3)) (A001353). - _G. C. Greubel_, Jul 10 2021
%o (Magma) [n le 4 select 1 else 4*Self(n-2) - Self(n-4): n in [1..41]]; // _G. C. Greubel_, Jul 10 2021
%o (Sage)
%o def a(n): return 1 if (n<5) else 4*a(n-2) - a(n-4)
%o [a(n) for n in (1..40)] # _G. C. Greubel_, Jul 10 2021
%Y Cf. A001353, A001835.
%K nonn
%O 1,5
%A _Roger L. Bagula_, Sep 17 2006
%E Edited by _G. C. Greubel_, Jul 10 2021