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Expansion of 1/(1 + x + x/(1 + x^2 + x^2/(1 + x^3 + x^3/(1 + x^4 + x^4/(1 + ...))))), a continued fraction.
1

%I #4 Sep 25 2017 13:08:30

%S 1,-2,4,-6,8,-12,18,-24,32,-44,58,-76,100,-128,164,-210,264,-332,416,

%T -516,640,-790,968,-1184,1444,-1752,2120,-2560,3078,-3692,4420,-5272,

%U 6276,-7456,8832,-10444,12326,-14512,17056,-20012,23432,-27392,31972,-37248,43332,-50338,58380,-67616,78208,-90328,104196

%N Expansion of 1/(1 + x + x/(1 + x^2 + x^2/(1 + x^3 + x^3/(1 + x^4 + x^4/(1 + ...))))), a continued fraction.

%t nmax = 50; CoefficientList[Series[1/(1 + x + ContinuedFractionK[x^k, 1 + x^(k + 1), {k, 1, nmax}]), {x, 0, nmax}], x]

%Y Cf. A088354.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Sep 25 2017