OFFSET
1,2
COMMENTS
Suppose that r is a real number in the interval [3/2, 5/3). Let C(r) = (c(k)) be the sequence of coefficients in the Maclaurin series for 1/(Sum_{k>=0} floor((k+1)*r))(-x)^k). It appears that c(k) > 0 for all k >= 0. Indeed, it appears that C(r) is strictly increasing and that the limit L(r) of c(k+1)/c(k) as k -> oo exists. Following is a guide for selected numbers r.
** r ** C(r) L(r)
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling, Another question about the golden ratio and other numbers, MathOverflow, Jan 17 2017.
FORMULA
a(n) = d*[w(n)*a(1)-w(n-1)*a(2)+...+d*w(2)*a(n-1)], where d=(-1)^n, with a(1)=1 and w=floor(n*tau), tau=(1+sqrt(5))/2.
EXAMPLE
a(5) = 17 = -[w(5)*a(1)-w(4)*a(2)+w(3)*a(3)-w(2)*a(4)] = -8*1+6*3-4*5+3*9. (a(1),a(2),...,a(n))(*)(w(1),-w(2),w(3),...,-d*w(n)) = (1,0,0,...,0), where (*) denotes convolution, w = lower Wythoff sequence, A000201.
MATHEMATICA
CoefficientList[Series[1/Sum[Floor[GoldenRatio*(k + 1)] (-x)^k, {k, 0, 50}],
{x, 0, 50}], x] (* Clark Kimberling, Dec 12 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Nov 23 2002
EXTENSIONS
Comments added by Clark Kimberling, Jul 10 2017
STATUS
approved