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A239466 Expansion of (1 - x + x^2 + sqrt(1 + 2*x + 3*x^2 - 2*x^3 + x^4)) / 2 in powers of x.

%I A239466 #11 Sep 08 2022 08:46:07
%S A239466 1,0,1,-1,1,0,-2,4,-3,-5,20,-29,1,94,-221,191,327,-1454,2282,-162,
%T A239466 -8002,19902,-18275,-30505,143511,-234364,24437,841723,-2164873,
%U A239466 2069014,3325410,-16315410,27375369,-3714435,-98829168,260605269,-257026289,-395719442
%N A239466 Expansion of (1 - x + x^2 + sqrt(1 + 2*x + 3*x^2 - 2*x^3 + x^4)) / 2 in powers of x.
%H A239466 G. C. Greubel, <a href="/A239466/b239466.txt">Table of n, a(n) for n = 0..1000</a>
%F A239466 G.f.: 1 - x + x^2 + x / (1 - x + x^2 + x / (1 - x + x^2 + x / ...)).  (continued fraction convergence is one power series term per iteration).
%F A239466 G.f.: 1 + x^2 / (1 + x / (1 + x^2 / (1 + x / ...))). (continued fraction convergence is three power series terms per iteration).
%F A239466 a(n) = - A129509(n) if n>2.
%F A239466 HANKEL transform is period 8 sequence A112299(n+5) = [1, 1, -1, 0, 1, -1, -1, 0, ...].
%F A239466 HANKEL transform of a(n+1) is period 8 sequence -A112299(n+4) = [0, -1, -1, 1, 0, -1, 1, 1, ...].
%F A239466 D-finite with recurrence: n*a(n) +(2*n-3)*a(n-1) +3*(n-3)*a(n-2) +(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2020
%e A239466 G.f. = 1 + x^2 - x^3 + x^4 - 2*x^6 + 4*x^7 - 3*x^8 - 5*x^9 + 20*x^10 + ...
%t A239466 CoefficientList[Series[(1-x+x^2 +Sqrt[1+2*x+3*x^2-2*x^3+x^4])/2, {x, 0, 50}], x] (* _G. C. Greubel_, Aug 08 2018 *)
%o A239466 (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - x + x^2 + sqrt(1 + 2*x + 3*x^2 - 2*x^3 + x^4 + x * O(x^n))) / 2, n))};
%o A239466 (PARI) {a(n) = my(A = 1 + O(x)); for(k=1, ceil(n / 3), A = 1 + x^2 / (1 + x / A)); polcoeff(A, n)};
%o A239466 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x+x^2 +Sqrt(1+2*x+3*x^2-2*x^3+x^4))/2)); // _G. C. Greubel_, Aug 08 2018
%Y A239466 Cf. A112299, A129509.
%K A239466 sign
%O A239466 0,7
%A A239466 _Michael Somos_, Mar 19 2014