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Expansion of the continued fraction 1 / (1-q / (1-q-q^2 / (1-q-q^2-q^3 / (1-q-q^2-q^3-q^4 / (...))))).
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%I #19 Jun 26 2022 09:31:55

%S 1,1,2,5,13,35,96,266,742,2079,5843,16457,46423,131099,370527,1047858,

%T 2964698,8390837,23754234,67260645,190478213,539484321,1528094423,

%U 4328632609,12262352881,34738763766,98416624789,278825903115,789961599608,2238129694407,6341171821627,17966261019890,50903653156245

%N Expansion of the continued fraction 1 / (1-q / (1-q-q^2 / (1-q-q^2-q^3 / (1-q-q^2-q^3-q^4 / (...))))).

%F a(n) ~ c * d^n, where d = 2.8333645039948803621658813720055524872119... and c = 0.1710130167563241590871776261530008679... - _Vaclav Kotesovec_, Jun 16 2022

%F a(n) = [q^n] K_{k>=0} -q^k / ((q^(k + 1) - 2*q + 1)/(q - 1)), where K is Gauss's notation for continued fractions. - _Peter Luschny_, Jun 20 2022

%t nmax = 40; CoefficientList[Series[1/(1 - x/(1 - x + ContinuedFractionK[-x^k, 1 - x*(1 - x^k)/(1 - x), {k, 2, nmax}])), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 16 2022 *)

%t A355040List[nmax_] := Module[{a, b, q},

%t a = ContinuedFractionK[-q^k, (q^(k + 1) - 2 q + 1)/(q - 1), {k, 0, nmax}];

%t b = Series[a, {q, 0, nmax}]; CoefficientList[b, q] ];

%t A355040List[32] (* _Peter Luschny_, Jun 20 2022 *)

%o (PARI) N=44; q='q+O('q^N);

%o f(n) = 1 - sum(k=1,n-1,q^k);

%o s=1; forstep(j=N, 1, -1, s = q^j/s; s = f(j) - s ); s = 1/s;

%o Vec(s)

%Y Cf. A230823, A355043, A355046.

%Y Cf. A088354.

%K nonn

%O 0,3

%A _Joerg Arndt_, Jun 16 2022