Sergei Duzhin - cp's OEIS frontend

User: Sergei Duzhin

Sergei Duzhin has authored 3 sequences.

A229555 An exotic continued fraction for the real root of 6y^3 + 4y^2 - 4y - 7.

1, 22, 1, 31, 2, 3, 1, 63, 1, 10, 1, 2, 1, 7, 1, 160905, 2, 1, 4, 58, 2, 2, 1, 2, 1, 7, 3, 1, 3, 1, 4, 3, 1, 47, 1, 214540, 1, 2, 9, 1, 45, 1, 3, 1, 48, 1, 21, 1, 9, 1, 8, 1, 2, 249610, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 20, 1, 4, 19, 1, 2, 1, 1, 1, 1, 3, 4, 1, 1, 1

Offset: 0

Sergei Duzhin, Oct 01 2013

Among its 148 initial numbers, this sequence has 8 unexpectedly big terms which are considerably bigger than those of the well-known exotic sequence A002937. It is peculiar that the ratio of the biggest of these denominators for the two examples, as well as the ratio of the smallest ones, is almost exactly 7: 115270760/16467250 = 7.000000607... and 160905/22986 = 7.000130514...
Deleting the first term in the sequence, we get a continued fraction for the algebraic integer given as the real root of the equation x^3 - 22*x^2 - 22*x - 6 = 0. It has the same set of big denominators, only shifted one position towards the beginning. - Sergei Duzhin, Oct 03 2013
For all integer cubic polynomials a*y^3 + b*y^2 + c*y + d with 0 < a <= 7, |b| <= 7, |c| <= 7, |d| <= 7, the given one (together with its three modifications obtained by a change of one sign and a palindromic reversal of coefficients) is the only polynomial in this set that, among its first 200 denominators, contains a number greater than 10^8, thus establishing a record. - Sergei Duzhin, Oct 04 2013

T. D. Noe, Table of n, a(n) for n = 0..1000

Digits:=500:
with(numtheory):
x:=fsolve(6*y^3 + 4*y^2 - 4*y - 7);
cfrac(x,200,'quotients');

r = Roots[6 x^3 + 4 x^2 - 4 x - 7 == 0, x][[1, 2]]; ContinuedFraction[r, 115] (* T. D. Noe, Oct 02 2013 *)

\p 250
contfrac(real(polroots(Pol([6,4,-4,-7]))[1])) \\ Charles R Greathouse IV, Oct 01 2013

y = ((2906 - 126*sqrt(3*163))^(1/3) + (2906 + 126*sqrt(3*163))^(1/3) - 4) / 18. - Andrey Zabolotskiy, Jan 21 2023

A128885 Number of closed orientable irreducible 3-manifolds of complexity n.

Original entry on oeis.org

3, 2, 4, 7, 14, 31, 74, 175, 436, 1154, 3078, 8421, 23434

Offset: 0

Sergei Duzhin, Apr 19 2007

S. Matveev, Algorithmic Topology and Classification of 3-Manifolds, Springer, 2003.

S. Matveev, Atlas of 3-manifolds.

Term 23434 added by Sergei Duzhin, Jan 28 2013

A054993 Number of "long curves", i.e., topological types of smooth embeddings of the oriented real line into the oriented plane that coincide with the standard immersion x -> (x,0) in the neighborhood of -infinity and +infinity.

Original entry on oeis.org

1, 2, 8, 42, 260, 1796, 13396, 105706, 870772, 7420836, 65004584, 582521748, 5320936416, 49402687392, 465189744448, 4434492302426, 42731740126228, 415736458808868, 4079436831493480, 40338413922226212, 401652846850965808, 4024556509468827432, 40558226664529024000, 410887438338905738908, 4182776248940752113344, 42770152711524569532616, 439143340987014152920384, 4526179842103708969039296

Offset: 0

Sergei Duzhin, Nov 11 2000

Also the number of knot diagrams with n crossings and two outgoing strings.

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
S. M. Gusein-Zade, On the enumeration of curves from infinity to infinity, in: Singularities and Bifurcations, Adv. Sov. Math., v. 21 (1994), pp. 189-198.

Steven R. Finch, Knots, links and tangles, August 8, 2003. [Cached copy, with permission of the author]
S. M. Gusein-Zade and F. S. Duzhin, On the number of topological types of plane curves; (Russian) Uspekhi Mat. Nauk 53 (1998), no. 3(321), 197-198. English translation: Russian Mathematical Surveys 53 (1998) 626-627. Related program and data.
J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Knots
J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Colored Links, J. Knot Theory, 10 (2001), 1233-1267.
Christoph Lamm, The enumeration of doubly symmetric diagrams for strongly positive amphicheiral knots, arXiv:2410.06601 [math.GT], 2024. See p. 14.
P. Zinn-Justin and J.-B. Zuber, Knot theory and matrix integrals, arXiv:1006.1812 [math-ph], 2010.
Index entries for sequences related to knots

Cf. A008980, A008981, A008982, A008983, A008984, A008985.
A151374 enumerates the long curves having Gauss diagrams without intersections, cf. A118814.
Cf. A067647, A067648.
A column of the triangles in A067640 and A062038.

Extended to n = 22 by J. L. Jacobsen and Paul Zinn-Justin, Jan 30 2002

More terms from Paul Zinn-Justin, Dec 13 2016

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User: Sergei Duzhin

A229555 An exotic continued fraction for the real root of 6y^3 + 4y^2 - 4y - 7.

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A128885 Number of closed orientable irreducible 3-manifolds of complexity n.

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A054993 Number of "long curves", i.e., topological types of smooth embeddings of the oriented real line into the oriented plane that coincide with the standard immersion x -> (x,0) in the neighborhood of -infinity and +infinity.

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