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 A206738 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. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS From Peter Bala, 15 Dec 2015: (Start) 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. (End) LINKS FORMULA a(n) ~ c * d^n, where d = 1.26326802855134275222... and c = 0.16506173508242936... - Vaclav Kotesovec, Aug 25 2017 From Peter Bala, Jul 03 2019 (Start) 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) EXAMPLE 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 MATHEMATICA 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 *) PROG (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), ", ")) CROSSREFS Cf. A206737, A143951, A005169, A111317, A143951, A224704. Sequence in context: A239948 A034391 A239243 * A282971 A174618 A144241 Adjacent sequences:  A206735 A206736 A206737 * A206739 A206740 A206741 KEYWORD nonn,easy AUTHOR Paul D. Hanna, Feb 12 2012 STATUS approved

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Last modified August 18 22:45 EDT 2022. Contains 356215 sequences. (Running on oeis4.)