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A206738
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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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2
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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, 116, 144, 184, 231, 292, 370, 465, 591, 742, 942, 1185, 1502, 1893, 2395, 3023, 3819, 4826, 6093, 7702, 9724, 12290, 15519, 19611, 24767, 31294, 39527, 49937, 63082
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OFFSET
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0,10
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) ~ c * d^n, where d = 1.26326802855134275222... and c = 0.16506173508242936... - Vaclav Kotesovec, Aug 25 2017
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.
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)
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EXAMPLE
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G.f.: A(x) = 1 + x^2 + x^4 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + 3*x^11 + ...
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
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MAPLE
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N:= 100:
C:= [0, [1, 1], seq([-x^i, 1], i=2..N, 3)]:
S:= series(numtheory:-cfrac(C), x, N+1):
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MATHEMATICA
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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 *)
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PROG
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(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)}
for(n=0, 55, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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