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%I #32 Feb 28 2024 14:09:48
%S 1,1,1,1,1,2,3,4,5,7,10,14,20,28,39,54,76,107,150,210,294,412,578,811,
%T 1137,1593,2233,3131,4390,6155,8629,12097,16959,23777,33336,46737,
%U 65524,91863,128790,180563,253149,354912,497581,697602,978031,1371190,1922395
%N G.f.: 1/(1 - x/(1 - x^4/(1 - x^7/(1 - x^10/(1 - x^13/(1 - x^16/(1 -...- x^(3*n-2)/(1 -...)))))))), a continued fraction.
%C We have the simple continued fraction expansions (A(x) is the sequence o.g.f.): A(1/n) = [1; n - 2, 1, n^3 - 2, 1, n^4 - 2, 1, n^6 - 2, 1, n^7 - 2, 1, n^9 - 2, 1, n^10 - 2, 1, ...] for n >= 3 and A(-1/n) = [0; 1, n - 1, 1, n^3 - 1, n^4 - 1, 1, n^6 - 1, n^7 - 1, 1, n^9 - 1, n^10 - 1, 1, ...] for n >= 2. Cf. A005169, A111317 and A143951. - _Peter Bala_, Dec 15 2015
%H Robert Israel, <a href="/A206737/b206737.txt">Table of n, a(n) for n = 0..2000</a>
%H Peter Bala, <a href="/A206737/a206737.pdf">Some simple continued fraction expansions associated with A206737</a>
%F a(n) ~ c * d^n, where d = 1.40198938377739909105003523518827... and c = 0.34165269320144328278000954698... - _Vaclav Kotesovec_, Aug 25 2017
%F From _Peter Bala_, Jul 03 2019: (Start)
%F O.g.f. as a ratio of q-series: N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2+n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))) and D(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2-2*n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))). Cf. A143951, A224704 and A206738.
%F D(q) has a simple real zero at x = 0.7132721628.... The constants c and d quoted in the above asymptotic approximation for a(n) are given by d = 1/x and c = - N(x)/(x*D'(x)), where the prime indicates differentiation w.r.t. q. (End)
%e G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ...
%e Simple continued fraction expansions: A(1/10) = 1.11112345816325284441923227158 ... = [1, 8, 1, 998, 1, 9998, 1, 999998, 1, 9999998, 1, 999999998, 1, 9999999998, 1, ...]; A(-1/10) = 0.909082643877542661578687284018 ... = [0, 1, 9, 1, 999, 9999, 1, 999999, 9999999, 1, 999999999, 9999999999, 1, ...]. - _Peter Bala_, Dec 15 2015
%p N:= 100: # to get a(0) .. a(N)
%p C:= [0,[1,1],seq([-x^i,1],i=1..N,3)]:
%p S:= series(numtheory:-cfrac(C),x,N+1):
%p seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Dec 28 2015
%t max = 15; CF = 1+x*O[x]^max; M = Sqrt[max+1]//Floor; For[k=0, k <= M, k++, CF = 1/(1-x^(3M-3k+1)*CF)]; CoefficientList[CF, x] (* _Jean-François Alcover_, Dec 29 2015, adapted from PARI *)
%t nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(3*Range[nmax + 1]-2)]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 25 2017 *)
%o (PARI) {a(n)=local(CF=1+x*O(x^n),M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^(3*M-3*k+1)*CF)); polcoeff(CF, n, x)}
%o for(n=0,55,print1(a(n),", "))
%Y Cf. A206738, A143951, A005169, A111317, A143951, A224704.
%K nonn
%O 0,6
%A _Paul D. Hanna_, Feb 12 2012
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