cp's OEIS Frontend

A078140 Convolutory inverse of signed lower Wythoff sequence.

%I A078140 #17 Jul 10 2017 22:50:01
%S A078140 1,3,5,9,17,30,52,90,154,262,446,758,1285,2176,3683,6230,10533,17803,
%T A078140 30085,50831,85873,145063,245037,413891,699082,1180761,1994293,
%U A078140 3368302,5688920,9608292,16227841,27407792,46289925,78180465,132041227
%N A078140 Convolutory inverse of signed lower Wythoff sequence.
%C A078140 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.
%C A078140 ** r **           C(r)       L(r)
%C A078140 sqrt(7/3)        A188135    A288238
%C A078140 Pi/2             A288229    A288239
%C A078140 sqrt(5/2)        A288230    A288240
%C A078140 4^(1/3)          A288231    A288241
%C A078140 (1 + sqrt(5))/2  A078140    A281112
%C A078140 3e/5             A288232    A288242
%C A078140 sqrt(8/3)        A288233    A288935
%C A078140 -1 + sqrt(7)     A288234    A289003
%C A078140 sqrt(e)          A288235    A289005
%C A078140 -4/5 + sqrt(6)   A288236    A289032
%C A078140 sqrt(11/4)       A288237    A289033
%H A078140 Clark Kimberling, <a href="/A078140/b078140.txt">Table of n, a(n) for n = 1..1000</a>
%H A078140 Clark Kimberling, <a href="http://mathoverflow.net/questions/259821/another-question-about-the-golden-ratio-and-other-numbers"> Another question about the golden ratio and other numbers</a>, MathOverflow, Jan 17 2017.
%F A078140 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.
%e A078140 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.
%t A078140 CoefficientList[Series[1/Sum[Floor[GoldenRatio*(k + 1)] (-x)^k, {k, 0, 50}],
%t A078140 {x, 0,50}], x]  (* _Clark Kimberling_, Dec 12 2016 *)
%Y A078140 Cf. A000201, A077607, A281112, A279676.
%K A078140 nonn
%O A078140 1,2
%A A078140 _Clark Kimberling_, Nov 23 2002
%E A078140 Comments added by _Clark Kimberling_, Jul 10 2017