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 A078140 Convolutory inverse of signed lower Wythoff sequence. 40
 1, 3, 5, 9, 17, 30, 52, 90, 154, 262, 446, 758, 1285, 2176, 3683, 6230, 10533, 17803, 30085, 50831, 85873, 145063, 245037, 413891, 699082, 1180761, 1994293, 3368302, 5688920, 9608292, 16227841, 27407792, 46289925, 78180465, 132041227 (list; graph; refs; listen; history; text; internal format)
 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) sqrt(7/3)        A188135    A288238 Pi/2             A288229    A288239 sqrt(5/2)        A288230    A288240 4^(1/3)          A288231    A288241 (1 + sqrt(5))/2  A078140    A281112 3e/5             A288232    A288242 sqrt(8/3)        A288233    A288935 -1 + sqrt(7)     A288234    A289003 sqrt(e)          A288235    A289005 -4/5 + sqrt(6)   A288236    A289032 sqrt(11/4)       A288237    A289033 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 Cf. A000201, A077607, A281112, A279676. Sequence in context: A298338 A018162 A077879 * A279780 A289260 A279595 Adjacent sequences:  A078137 A078138 A078139 * A078141 A078142 A078143 KEYWORD nonn AUTHOR Clark Kimberling, Nov 23 2002 EXTENSIONS Comments added by Clark Kimberling, Jul 10 2017 STATUS approved

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Last modified January 24 21:06 EST 2021. Contains 340411 sequences. (Running on oeis4.)