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%I #25 Aug 23 2018 03:08:12
%S 1,-1,0,-1,0,-1,-1,-1,-2,-2,-4,-4,-7,-9,-13,-19,-25,-38,-51,-75,-104,
%T -149,-211,-298,-426,-600,-857,-1211,-1724,-2444,-3471,-4930,-6995,
%U -9940,-14104,-20038,-28444,-40397,-57362,-81453,-115675,-164250,-233262,-331227
%N Expansion of 1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - x^6/(1 - ... - x^n/(1 - ...))))))), a continued fraction.
%C The sequence b(n>=1) = 1, 0, 1, 0, 1, 1, 1, 2, 2, 4, 4, 7, ... of absolute values counts fountains of n coins that cannot be separated into two or more fountains by cutting vertically through the fountain without splitting a coin. (This separation requires that the fountain is a left-right sequence of more elementary fountains counted by b(n).) A005169(n) = Sum_{compositions n=n1+n2+n3+...} Product b(n1)*b(n2)*.... - _R. J. Mathar_, Aug 22 2018
%H Seiichi Manyama, <a href="/A291148/b291148.txt">Table of n, a(n) for n = 0..1000</a>
%F Convolution inverse of A005169.
%F a(n) ~ c * d^n, where d = 1.42009048763893649946106129818306075366296460727614... and c = -0.093433697175825717154301151812109730023054876584907211486145769... - _Vaclav Kotesovec_, Oct 16 2017
%e G.f. = 1 - x - x^3 - x^5 - x^6 - x^7 - 2*x^8 - 2*x^9 - ...
%Y Cf. A005169, A290975, A291147.
%K sign
%O 0,9
%A _Seiichi Manyama_, Aug 18 2017
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