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A217233 Expansion of (1-2*x+x^2)/(1-3*x-3*x^2+x^3).

%I A217233 #34 May 31 2025 05:17:49
%S A217233 1,1,7,23,89,329,1231,4591,17137,63953,238679,890759,3324361,12406681,
%T A217233 46302367,172802783,644908769,2406832289,8982420391,33522849271,
%U A217233 125108976697,466913057513,1742543253359,6503259955919,24270496570321,90578726325361
%N A217233 Expansion of (1-2*x+x^2)/(1-3*x-3*x^2+x^3).
%C A217233 Numbers with the property a(n)^2+a(n-1)^2 = 2*(a(n)-a(n-1)-(-1)^n)^2.
%H A217233 Vincenzo Librandi, <a href="/A217233/b217233.txt">Table of n, a(n) for n = 0..1000</a>
%H A217233 R. C. Alperin, <a href="https://www.fq.math.ca/Papers/57-4/alperin07132019.pdf">A family of nonlinear recurrences and their linear solutions</a>, Fib. Q., 57:4 (2019), 318-321.
%H A217233 R. C. Alperin, <a href="https://www.fq.math.ca/Papers/58-2/alperin09212019.pdf">A nonlinear recurrence and its relations to Chebyshev polynomials</a>, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
%H A217233 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-1).
%F A217233 G.f.: (1-x)^2/((1+x)*(1-4*x+x^2)).
%F A217233 a(n) = (4*(-2)^n+(1-sqrt(3))^(2*n+1)+(1+sqrt(3))^(2*n+1))/(6*2^n).
%F A217233 a(n) = -a(-n-1) = 3*a(n-1)+3*a(n-2)-a(n-3) = 4*a(n-1)-a(n-2)+4*(-1)^n.
%F A217233 a(n)+a(n-1) = A052530(n) with a(-1)=-1.
%F A217233 a(n)-a(n-2) = A003699(n) with n>1.
%F A217233 Sum(a(i), i=0..n) = A006253(n).
%e A217233 a(3)=23, a(2)=7: 23^2+7^2 = 2*(23-7-(-1)^3)^2 = 578;
%e A217233 a(6)=1231, a(5)=329: 1231^2+329^2 = 2*(1231-329-(-1)^6)^2 = 1623602.
%t A217233 CoefficientList[Series[(1 - 2 x + x^2)/(1 - 3 x - 3 x^2 + x^3), {x, 0, 25}], x]
%o A217233 (PARI) Vec((1-2*x+x^2)/(1-3*x-3*x^2+x^3)+O(x^26))
%o A217233 (Magma) m:=26; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x+x^2)/(1-3*x-3*x^2+x^3)));
%o A217233 (Maxima) makelist(coeff(taylor((1-2*x+x^2)/(1-3*x-3*x^2+x^3), x, 0, n), x, n), n, 0, 25);
%Y A217233 Cf. A109437 (1/(1-3*x-3*x^2+x^3)), A006253 ((1-x)/(1-3*x-3*x^2+x^3)).
%K A217233 nonn,easy
%O A217233 0,3
%A A217233 _Bruno Berselli_, Sep 28 2012