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A271929
G.f. A(x) satisfies: A(x)^3 = A(x^3) / (1 - 3*x).
4
1, 1, 2, 5, 12, 31, 83, 224, 615, 1708, 4777, 13455, 38110, 108428, 309714, 887666, 2551575, 7353423, 21240460, 61478489, 178269670, 517784717, 1506162369, 4387201004, 12795170784, 37359689295, 109199349181, 319493390481, 935616592227, 2742209152877, 8043500169958, 23610710680582, 69354125493930, 203852682699869, 599549063015417, 1764338532368820
OFFSET
1,3
COMMENTS
Compare g.f. to: G(x)^2 = G(x^2)/(1 - 2*x) where G(x) is the g.f. of A123916, the EULER transform of A000048.
LINKS
FORMULA
The EULER transform of A046211, where A046211(n) is the number of ternary Lyndon words whose digits sum to 1 (or 2) mod 3.
a(n) ~ c * 3^n / n^(2/3), where c = 0.1260671867244258410294918... . - Vaclav Kotesovec, Apr 18 2016
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 31*x^6 + 83*x^7 + 224*x^8 + 615*x^9 + 1708*x^10 + 4777*x^11 + 13455*x^12 +...
where A(x)^3 = A(x^3) / (1 - 3*x).
Also, when expressed as the EULER transform of A046211,
A(x) = x/( (1-x) * (1-x^2) * (1-x^3)^3 * (1-x^4)^6 * (1-x^5)^16 * (1-x^6)^39 * (1-x^7)^104 * (1-x^8)^270 * (1-x^9)^729 *...* (1-x^n)^A046211(n) *...).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 84*x^7 + 252*x^8 + 758*x^9 + 2274*x^10 + 6822*x^11 + 20471*x^12 + 61413*x^13 + 184239*x^14 +...
PROG
(PARI) {a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3)/(1 - 3*x +x*O(x^n)))^(1/3)); polcoeff(G=A, n)}
for(n=1, 50, print1(a(n), ", "))
CROSSREFS
Cf. A123916.
Sequence in context: A076906 A097893 A093379 * A071359 A160999 A014329
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 17 2016
STATUS
approved