OFFSET
0,6
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (-1 - x + x^2 + sqrt(1 + 2*x + 3*x^2 - 6*x^3 + x^4)) / (2 * (x - x^2)).
G.f.: x / (1 + x - x^2 + (x - x^2) * x / (1 + x - x^2 + (x - x^2) * x / ...)) continued fraction.
a(n) = A245734(-n) for all n in Z.
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.
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.
EXAMPLE
G.f. = x - x^2 + x^3 + x^4 - 5*x^5 + 9*x^6 - 3*x^7 - 29*x^8 + 89*x^9 + ...
MATHEMATICA
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 *)
PROG
(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)) };
(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))};
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 30 2014
STATUS
approved