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%I #18 Nov 03 2016 11:10:43
%S 1,1,1,2,2,3,5,6,10,14,21,32,46,71,104,157,235,350,527,785,1179,1763,
%T 2639,3954,5915,8861,13262,19857,29731,44507,66640,99765,149366,
%U 223625,334795,501247,750434,1123518,1682076,2518314,3770306,5644701,8450977,12652376
%N G.f.: 1/(1 - x*(1-x^3)/(1 - x^2*(1-x^4)/(1 - x^3*(1-x^5)/(1 - x^4*(1-x^6)/(1 - ...))))), a continued fraction.
%C Compare to the continued fraction representation for the g.f. of A173173, where A173173(n) = ceiling(Fibonacci(n)/2).
%C Limit a(n)/a(n+1) = 0.6679357039724580760720733281356826861233293827578332775311...
%H Alois P. Heinz, <a href="/A227360/b227360.txt">Table of n, a(n) for n = 0..1000</a>
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 10*x^8 +...
%t nMax = 44; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x] &; A227360 = col[2][[1 ;; nMax]] (* _Jean-François Alcover_, Nov 03 2016 *)
%o (PARI) {a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+3))*CF+x*O(x^n))); polcoeff(CF, n)}
%o for(n=0,50,print1(a(n),", "))
%Y Cf. A173173, A227374, A227375, A228644, A228645.
%Y Column m=2 of A185646.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jul 08 2013
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