A239466 - OEIS

%I #11 Sep 08 2022 08:46:07

%S 1,0,1,-1,1,0,-2,4,-3,-5,20,-29,1,94,-221,191,327,-1454,2282,-162,

%T -8002,19902,-18275,-30505,143511,-234364,24437,841723,-2164873,

%U 2069014,3325410,-16315410,27375369,-3714435,-98829168,260605269,-257026289,-395719442

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

%H G. C. Greubel, <a href="/A239466/b239466.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 G.f.: 1 + x^2 / (1 + x / (1 + x^2 / (1 + x / ...))). (continued fraction convergence is three power series terms per iteration).

%F a(n) = - A129509(n) if n>2.

%F HANKEL transform is period 8 sequence A112299(n+5) = [1, 1, -1, 0, 1, -1, -1, 0, ...].

%F HANKEL transform of a(n+1) is period 8 sequence -A112299(n+4) = [0, -1, -1, 1, 0, -1, 1, 1, ...].

%F 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 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 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 (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 (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 (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 Cf. A112299, A129509.

%K sign

%O 0,7

%A _Michael Somos_, Mar 19 2014