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A026106
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Number of polyhexes of class PF2 (with one catafusene annealated to pyrene).
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9
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2, 5, 16, 55, 208, 817, 3336, 13935, 59406, 257079, 1126948, 4992421, 22318048, 100546543, 456055730, 2080872845, 9544572590, 43984730855, 203550840696, 945562887981, 4407586685688, 20609668887723, 96646196091276, 454402001079165
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OFFSET
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5,1
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COMMENTS
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See reference for precise definition.
In Cyvin et al. (1992), sequence (N(m): m >= 1) = (A002212(m): m >= 1) is defined by eq. (1), p. 533. (We may let N(0) := A002212(0) = 1.)
Sequence (M(m): m >= 1) is defined by eq. (13), p. 534. We have M(2*m) = M(2*m-1) = A007317(m) for m >= 1.
Sequences (N(m): m >= 1) and (M(m): m >= 1) appear in Table 1, p. 533.
The current sequence is denoted by 1^Q_(4+n) (with n = 1,2,3,...). Thus, a(n+4) = 1^Q_(4+n) for n >= 1; i.e., a(m) = 1^Q_{m} for m >= 5. We have 1^Q_(4+n) = (1/2)*(3*N(n) + M(n)) for n >= 1. See eq. (33), p. 536.
Sequence (1^Q_(4+n): n >= 1) appears in Table II, p. 537.
We may use the many formulae in the documentations of sequences A002212 and A007317 in order to create complicated formulae and recurrence relations for (a(n): n >= 5). We omit the details.
The first g.f. below is a combination of the g.f. for sequence A002212 by John W. Layman in 2001 and the g.f. for sequence A007317 by Ira M. Gessel and Jang Soo Kim in 2010.
The second g.f. appears in eq. (A1), p. 1180, in Cyvin et al. (1994). It is algebraically equivalent to the first g.f.
(Apparently, the word "annealated" in Cyvin et al. (1992) is spelled "annelated" in Cyvin et al. (1994).)
(End)
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LINKS
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Eric Weisstein's World of Mathematics, Fusenes.
Eric Weisstein's World of Mathematics, Polyhex.
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FORMULA
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G.f.: (x^3/4)*(4 - 8*x - 3*sqrt(1 - 6*x + 5*x^2) - (x + 1)*sqrt((1 - 5*x^2)/(1 - x^2))).
G.f.: x^3*(1 - 2*x) - (x^3/4)*(3*(1 - x)^(1/2)*(1 - 5*x)^(1/2) + (1 - x)^(-1)*(1 - x^2)^(1/2)*(1 - 5*x^2)^(1/2)) (see eq. (A1), p. 1180, in Cyvin et al. (1994)).
(End)
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MAPLE
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bb := proc(x) (1/4)*x^3*(4-8*x-3*sqrt((1-x)*(1-5*x))-(x+1)*sqrt((1-5*x^2)/(1-x^2))) end proc;
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MATHEMATICA
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(1/4) x^3 (4 - 8x - 3Sqrt[(1-x)(1-5x)] - (x+1) Sqrt[(1-5x^2)/(1-x^2)]) + O[x]^29 // CoefficientList[#, x]& // Drop[#, 5]& (* Jean-François Alcover, Apr 24 2020, from Maple *)
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CROSSREFS
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Cf. A002212, A007317, A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534, A039658.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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