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A285903 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))).

%I A285903 #22 Feb 16 2025 08:33:44
%S A285903 1,0,1,2,4,7,14,23,43,73,134,223,405,689,1216,2094,3678,6333,11080,
%T A285903 19152,33363,57798,100549,174262,302898,525328,912448,1583069,2748892,
%U A285903 4769842,8281087,14371045,24946819,43295806,75153267,130434130,226401111,392944875,682038592,1183770424,2054659668,3566162246
%N A285903 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))).
%H A285903 Vaclav Kotesovec, <a href="/A285903/b285903.txt">Table of n, a(n) for n = 1..1000</a>
%H A285903 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%H A285903 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A285903 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = (Sum_{n>=0} (-1)^n*x^(n*(n+1)) /Product_{k=1..n} (1 - x^k)) / (Sum_{n>=0} (-1)^n*x^(n^2)/Product_{k=1..n} (1 - x^k)).
%F A285903 Sum_{d|n} a(d) = A005169(n) for n > 0.
%F A285903 From _Vaclav Kotesovec_, Apr 30 2017: (Start)
%F A285903 a(n) ~ c * d^n, where
%F A285903 d = 1/A347901 = 1.735662824530347425658260749719668530254652847290392754609934...
%F A285903 c = 0.31236332459674145306627970724066492149823012868471473538681348971946...
%F A285903 (End)
%e A285903 G.f.: 1 + x/(1 - x) + x^3/(1 - x^3) + 2*x^4/(1 - x^4) + 4*x^5/(1 - x^5) + ... = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))).
%t A285903 nn = 42; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-x^n, 1, {n, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
%Y A285903 Cf. A005169, A192728.
%K A285903 nonn
%O A285903 1,4
%A A285903 _Ilya Gutkovskiy_, Apr 28 2017