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A160254 Arising from lower and upper bounds on the number of numerical semigroups of genus n. 1
1, 2, 4, 7, 13, 24, 44, 81, 151, 280, 525, 984, 1859, 3511, 6682, 12709, 24334, 46565, 89626, 172381, 333262, 643733, 1249147, 2421592, 4713715, 9165792, 17888456, 34873456, 68212220, 133269997, 261167821, 511211652, 1003436520, 1967293902 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Table 1, p.8 of Elizalde. A000045(n-2) = F(n-2) <= A007323(n) <= a(n) <= 1+3*(2^(n-3))). Abstract: We improve the previously best known lower and upper bounds on the number n_g of numerical semigroups of genus g. Starting from a known recursive description of the tree T of numerical semigroups, we analyze some of its properties and use them to construct approximations of T by generating trees whose nodes are labeled by certain parameters of the semigroups. We then translate the succession rules of these trees into functional equations for the generating functions that enumerate their nodes, and solve these equations to obtain the bounds. Some of our bounds involve the Fibonacci numbers, and the others are expressed as generating functions. We also give upper bounds on the number of numerical semigroups having an infinite number of descendants in T.

LINKS

Matthew House, Table of n, a(n) for n = 1..3328

Sergi Elizalde, Improved bounds on the number of numerical semigroups of a given genus, arXiv:0905.0489 [math.CO], May 4, 2009.

FORMULA

G.f.: x*(2-3*x+x^2-4*x^3+3*x^4-2*x^5+x*(1-x-x^3)*sqrt((1+2x)/(1-2x)))/(2(1-3*x+3*x^2-3*x^3+4*x^4-3*x^5+2*x^6)).

CROSSREFS

Cf. A000045, A007323.

Sequence in context: A054175 A000073 A255069 * A276661 A005318 A102111

Adjacent sequences:  A160251 A160252 A160253 * A160255 A160256 A160257

KEYWORD

nonn,uned

AUTHOR

Jonathan Vos Post, May 06 2009

STATUS

approved

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Last modified November 18 01:16 EST 2017. Contains 294837 sequences.