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G.f.: 1/(1 - x^2/(1 - x^5/(1 - x^8/(1 - x^11/(1 - x^14/(1 - x^17/(1 -...- x^(3*n-1)/(1 -...)))))))), a continued fraction.
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%I #31 Feb 28 2024 14:08:25

%S 1,0,1,0,1,0,1,1,1,2,1,3,2,4,4,6,7,9,11,14,18,22,29,35,46,56,73,90,

%T 116,144,184,231,292,370,465,591,742,942,1185,1502,1893,2395,3023,

%U 3819,4826,6093,7702,9724,12290,15519,19611,24767,31294,39527,49937,63082

%N G.f.: 1/(1 - x^2/(1 - x^5/(1 - x^8/(1 - x^11/(1 - x^14/(1 - x^17/(1 -...- x^(3*n-1)/(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 - 2, 1, n^3 - 2, 1, n^5 - 2, 1, n^6 - 2, 1, n^8 - 2, 1, n^9 - 2, 1, n^11 - 2, 1, n^12 - 2, 1, ...] for n >= 2 and A(-1/n) = [ 1, n^2 - 1, n^3 - 1, 1, n^5 - 1, n^6 - 1, 1, n^8 - 1, n^9 - 1, 1, n^11 - 1, n^12 - 1, 1, ...] for n >= 2. Cf. A005169, A111317 and A143951. - _Peter Bala_, Dec 15 2015

%H Robert Israel, <a href="/A206738/b206738.txt">Table of n, a(n) for n = 0..2000</a>

%F a(n) ~ c * d^n, where d = 1.26326802855134275222... and c = 0.16506173508242936... - _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+2*n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))) and D(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2-n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))). Cf. A143951, A224704 and A206737.

%F D(q) has a simple real zero at x = 0.79159764784576529644 .... The constants c and d quoted in the above asymptotic approximation 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^2 + x^4 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + 3*x^11 + ...

%e Simple continued fraction expansions: A(1/2) = 1.34788543155288690684 ... = [1; 2, 1, 6, 1, 30, 1, 62, 1, 254, 1, 510, 1, 2046, 1, 4094, 1, ...] and A(-1/2) = 1.3199498363818812865 ... = [1; 3, 7, 1, 31, 63, 1, 255, 511, 1, 2047, 4095, 1, ...]. - _Peter Bala_, Dec 15 2015

%p N:= 100:

%p C:= [0,[1,1],seq([-x^i,1],i=2..N,3)]:

%p S:= series(numtheory:-cfrac(C),x,N+1):

%p seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Feb 18 2024

%t nmax = 60; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(3*Range[nmax + 1]-1)]], {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+2)*CF)); polcoeff(CF, n, x)}

%o for(n=0,55,print1(a(n),", "))

%Y Cf. A206737, A143951, A005169, A111317, A143951, A224704.

%K nonn,easy

%O 0,10

%A _Paul D. Hanna_, Feb 12 2012