A245735 - OEIS

%I #12 Sep 08 2022 08:46:09

%S 0,1,-1,1,1,-5,9,-3,-29,89,-107,-121,833,-1703,631,6705,-21943,27587,

%T 34937,-242427,507739,-172615,-2201619,7253775,-9083263,-12931023,

%U 86757487,-181330015,52436881,843605643,-2751023447,3373429837,5393254483,-34585122919

%N G.f. A(x) satisfies 0 = A(0) and 0 = f(x, A(x)) where f(u, v) = (u - v) * (1 + u*v) - u*v * (1 - u*v).

%H G. C. Greubel, <a href="/A245735/b245735.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (-1 - x + x^2 + sqrt(1 + 2*x + 3*x^2 - 6*x^3 + x^4)) / (2 * (x - x^2)).

%F G.f.: x / (1 + x - x^2 + (x - x^2) * x / (1 + x - x^2 + (x - x^2) * x / ...)) continued fraction.

%F a(n) = A245734(-n) for all n in Z.

%F 0 = +a(n)*(n) +a(n+1)*(-7*n-9) +a(n+2)*(9*n+18) +a(n+3)*(-n) +a(n+4)*(-n-3) +a(n+5)*(-n-6) for all n in Z.

%F 0 = a(n)*(+a(n+1) -16*a(n+2) +27*a(n+3) -a(n+4) -4*a(n+5) -7*a(n+6)) + a(n+1)*(+2*a(n+1) +40*a(n+2) -109*a(n+3) +24*a(n+5) +40*a(n+6)) + a(n+2)*(-18*a(n+2) +97*a(n+3) +4*a(n+4) -44*a(n+5) -45*a(n+6)) +a(n+3)*(-27*a(n+3) +a(n+4) +31*a(n+5) +7*a(n+6)) +a(n+4)*(-2*a(n+4) -4*a(n+5) +4*a(n+6)) +a(n+5)*(-2*a(n+5) +a(n+6)) for all n in Z.

%e G.f. = x - x^2 + x^3 + x^4 - 5*x^5 + 9*x^6 - 3*x^7 - 29*x^8 + 89*x^9 + ...

%t CoefficientList[Series[(-1 - x + x^2 + Sqrt[1 + 2*x + 3*x^2 - 6*x^3 + x^4])/(2*(x - x^2)), {x, 0, 60}], x] (* _G. C. Greubel_, Aug 06 2018 *)

%o (PARI) {a(n) = my(A); A = O(x); if( n<0, for(k=1, -n, A = x / (1 - (x + x^2) - (1 - x) * A)), for(k=0, n/2, A = x / (1 + (x - x^2) + (x - x^2) * A))); polcoeff(A, abs(n)) };

%o (PARI) {a(n) = polcoeff( if( n<0, ((1 - x - x^2) - sqrt(1 - 6*x + 3*x^2 + 2*x^3 + x^4 + x * O(x^-n) )) / (2 * (1 - x)), ((-1 - x + x^2) + sqrt(1 + 2*x + 3*x^2 - 6*x^3 + x^4 + x^2 * O(x^n) )) / (2 * (x - x^2))), abs(n))};

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((-1 - x +x^2 +Sqrt(1+2*x+3*x^2-6*x^3+x^4))/(2*(x-x^2))));  // _G. C. Greubel_, Aug 06 2018

%Y Cf. A245734.

%K sign

%O 0,6

%A _Michael Somos_, Jul 30 2014