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A047051 Prime alternating tangle types (of knots) with n crossings. 1
1, 2, 4, 10, 29, 98, 372, 1538, 6755, 30996, 146982, 715120, 3552254, 17951322, 92045058, 477882876, 2508122859, 13289437362, 71010166670, 382291606570, 2072025828101, 11298920776704, 61954857579594, 341427364138880 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

G.f. is related to the classes of 2- and 3-connected planar maps with n edges. Further terms are known.

REFERENCES

C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacif. J. Math., 182, No 2 (1998), 329-358.

LINKS

Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007, Table of n, a(n) for n = 1..50

S. R. Finch, Knots, links and tangles

P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, p. 12.

P. Zinn-Justin, The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links

C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacif. J. Math., 182, No 2 (1998), 329-358.

MATHEMATICA

max = 24; Clear[a, eq, s]; gf = Sum[a[k]*x^k, {k, 0, max}]; a[0] = 0; a[1] = 1; a[2] = 2; coes = CoefficientList[(x^4 - 2*x^3 + x^2)*gf^5 + (8*x^4 - 14*x^3 + 8*x^2 - 2*x)*gf^4 + (25*x^4 - 16*x^3 - 14*x^2 + 8*x + 1)*gf^3 + (38*x^4 + 15*x^3 - 30*x^2 - x + 2)*gf^2 + (28*x^4 + 36*x^3 - 5*x^2 - 12*x + 1)*gf + 8*x^4 + 17*x^3 + 8*x^2 - x, x]; eq[n_] := eq[n] = If[n == 1, Thread[Drop[coes, 3] == 0], eq[n-1] /. s[n-1] // First]; s[n_] := s[n] = (Print["n = ", n]; Solve[eq[n][[n]], a[n+2]]); sol = Table[s[n], {n, 1, max-2}] // Flatten; Table[a[n], {n, 1, max}] /. sol (* Jean-Fran├žois Alcover, Apr 15 2014 *)

CROSSREFS

Cf. A002864, A000139, A000287.

Sequence in context: A230957 A279552 A261041 * A271077 A126349 A076315

Adjacent sequences:  A047048 A047049 A047050 * A047052 A047053 A047054

KEYWORD

easy,nice,nonn

AUTHOR

Valery A. Liskovets

EXTENSIONS

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

STATUS

approved

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Last modified November 15 11:35 EST 2018. Contains 317238 sequences. (Running on oeis4.)