A351488 -id:A351488 - cp's OEIS frontend

A082640 Triangle T(m,n) read by rows: unimodular triangulations of the grid P(m,n), m,n > 0, n <= m.

2, 6, 64, 20, 852, 46456, 70, 12170, 2822648, 736983568, 252, 182132, 182881520, 208902766788, 260420548144996, 924, 2801708, 12244184472, 61756221742966, 341816489625522032, 1999206934751133055518, 3432, 43936824, 839660660268, 18792896208387012, 464476385680935656240, 12169409954141988707186052, 332633840844113103751597995920

Offset: 1

Ralf Stephan, May 15 2003

The limit of T(2,n)^(1/n) is (611+sqrt(73))/36. - Stepan Orevkov, Jan 31 2022

			Triangle begins:
    2;
    6,     64;
   20,    852,     46456;
   70,  12170,   2822648,    736983568;
  252, 182132, 182881520, 208902766788, 260420548144996;
  ...

Stepan Orevkov, Table of n, a(n) for n = 1..55 (rows 1 to 10).
V. Kaibel and G. M. Ziegler, Counting Lattice Triangulations, arXiv:math/0211268 [math.CO], 2002.
S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022.
S. Yu. Orevkov, Asymptotics of the number of lattice triangulations of rectangles of width 4 and 5, arXiv:2412.17065 [math.CO], 2024.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.

Row/columns 1..2 are A000984, A296165.
Row/columns 5..9 are A351484, A351485, A351486, A351487, A351488.
Row sums: A151686. - N. J. A. Sloane, Jun 02 2009

A351484 a(n) is the number of primitive triangulations of a 5 X n rectangle.

Original entry on oeis.org

252, 182132, 182881520, 208902766788, 260420548144996, 341816489625522032, 464476385680935656240, 645855159466371391947660, 913036902513499041820702784, 1306520849733616781789190513820, 1887591165891651253904039432371172, 2747848427721241461905176361078147168

Offset: 1

Stefano Spezia, Feb 12 2022

nonn

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Table 2, p. 6.

Cf. A082640 (m X n rectangle).
Cf. A351480, A351481, A351482, A351483.
Cf. A351485, A351486, A351487, A351488.

A351485 a(n) is the number of primitive triangulations of a 6 X n rectangle.

Original entry on oeis.org

924, 2801708, 12244184472, 61756221742966, 341816489625522032, 1999206934751133055518, 12169409954141988707186052, 76083336332947513655554918994, 484772512167266688498399632918196, 3131521959869770128138491287826065904, 20443767611927599823217291769468449488548

Offset: 1

Stefano Spezia, Feb 12 2022

nonn

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Table 3, p. 7.

Cf. A082640 (m X n rectangle).
Cf. A351480, A351481, A351482, A351483.
Cf. A351484, A351486, A351487, A351488.

A351486 a(n) is the number of primitive triangulations of a 7 X n rectangle.

Original entry on oeis.org

3432, 43936824, 839660660268, 18792896208387012, 464476385680935656240, 12169409954141988707186052, 332633840844113103751597995920, 9369363517501208819530429967280708, 269621109753732518252493257828413137272, 7880009979020501614060394747170100093057300, 233031642883906149386619647304562977586311372556

Offset: 1

Stefano Spezia, Feb 12 2022

nonn

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Table 4, p. 7.

Cf. A082640 (m X n rectangle).
Cf. A351480, A351481, A351482, A351483.
Cf. A351484, A351485, A351487, A351488.

A351487 a(n) is the number of primitive triangulations of an 8 X n rectangle.

Original entry on oeis.org

12870, 698607816, 58591381296256, 5831528022482629710, 645855159466371391947660, 76083336332947513655554918994, 9369363517501208819530429967280708, 1191064812882685539785713745400934044308, 155023302820254133629368881178138076738462112, 20527337238769032315796332007167102984745417344046

Offset: 1

Stefano Spezia, Feb 12 2022

nonn

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Table 5, p. 8.

Cf. A082640 (m X n rectangle).
Cf. A351480, A351481, A351482, A351483.
Cf. A351484, A351485, A351486, A351488.

A351480 Decimal expansion of (611 + sqrt(73))/36.

Original entry on oeis.org

1, 7, 2, 0, 9, 5, 5, 5, 6, 5, 9, 5, 9, 2, 1, 5, 3, 6, 4, 3, 5, 5, 1, 9, 9, 0, 2, 3, 1, 2, 8, 4, 4, 3, 6, 2, 8, 9, 8, 4, 9, 8, 4, 5, 9, 8, 1, 3, 7, 5, 9, 2, 4, 5, 0, 6, 7, 1, 9, 6, 8, 4, 7, 5, 7, 0, 4, 9, 2, 1, 2, 4, 6, 7, 2, 0, 3, 5, 3, 6, 0, 6, 6, 1, 4, 1, 1, 3, 8, 1

Offset: 2

Stefano Spezia, Feb 12 2022

			17.2095556595921536435519902312844...

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Theorem 1, p. 2.
Index entries for algebraic numbers, degree 2

Cf. A010525, A082640.
Cf. A351481, A351482, A351483.
Cf. A351484, A351485, A351486, A351487, A351488.

First[RealDigits[(611+Sqrt[73])/36,10,90]]

Equals lim_{n->oo} A082640(2, n)^(1/n).

Equals 288*x_2, where x_2 is the largest root of 5184*x^2 - 611*x + 18.

A351481 Decimal expansion of log_2((611 + sqrt(73))/36)/2.

Original entry on oeis.org

2, 0, 5, 2, 5, 6, 8, 9, 7, 1, 6, 1, 2, 7, 3, 5, 6, 6, 5, 1, 0, 7, 8, 7, 1, 5, 4, 0, 4, 7, 8, 6, 5, 5, 8, 7, 1, 0, 5, 3, 8, 4, 8, 7, 6, 2, 3, 7, 1, 2, 2, 1, 4, 3, 8, 8, 9, 2, 9, 8, 0, 3, 2, 7, 7, 4, 1, 7, 9, 0, 8, 2, 0, 0, 4, 1, 2, 0, 7, 1, 0, 4, 6, 5, 9, 3, 2, 3, 6, 3

Offset: 1

Stefano Spezia, Feb 12 2022

			2.052568971612735665107871540478655871...

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Theorem 1, p. 2.
Index entries for transcendental numbers

Cf. A010525, A082640.
Cf. A351480, A351482, A351483.
Cf. A351484, A351485, A351486, A351487, A351488.

First[RealDigits[N[Log[2,(611+Sqrt[73])/36]/2,90]]]

log((611 + sqrt(73))/36)/log(4) \\ Charles R Greathouse IV, Oct 31 2023

Equals log_2(alpha)/2, where alpha = lim_{n->oo} A082640(2, n)^(1/n).

A351482 Decimal expansion of lim_{n->oo} f(3, n)^(1/(3*n)), where f(m, n) is the number of primitive lattice triangulations of m X n rectangle.

Original entry on oeis.org

4, 2, 3, 9, 3, 6, 9, 4, 8, 1, 5, 4, 8, 0, 2, 5, 6, 7, 1, 8, 7, 7, 6, 2, 5, 7, 4, 2, 0, 4, 5, 2, 3, 5, 7, 7, 2, 1, 0, 0, 6, 9, 5, 7, 1, 1, 2, 5, 1, 7, 9, 5, 4, 9, 9, 8, 3, 0, 8, 0, 1, 6, 8, 7, 8, 3, 3, 3, 5, 8, 2, 3, 8, 2, 7, 6, 7, 2, 8, 9, 8, 7, 8, 3, 7, 0, 5, 4, 8

Offset: 1

Stefano Spezia, Feb 12 2022

			4.2393694815480256718776257420452357721...

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Theorem 2, p. 2.

Cf. A082640.
Cf. A351480, A351481, A351483.
Cf. A351484, A351485, A351486, A351487, A351488.

A351483 Decimal expansion of log_2(lim_{n->oo} f(3, n)^(1/(3*n))), where f(m, n) is the number of primitive lattice triangulations of m X n rectangle.

Original entry on oeis.org

2, 0, 8, 3, 8, 4, 9, 7, 0, 9, 7, 2, 1, 0, 2, 3, 2, 0, 8, 2, 2, 4, 2, 1, 9, 2, 8, 9, 4, 9, 6, 1, 7, 0, 1, 3, 9, 6, 4, 8, 5, 1, 3, 4, 2, 3, 2, 4, 9, 5, 5, 2, 1, 3, 0, 7, 9, 9, 0, 5, 9, 9, 1, 8, 5, 5, 1, 7, 2, 9, 0, 6, 9, 2, 8, 1, 8, 0, 5, 2, 5, 1, 8, 6, 6, 5, 0, 8, 8

Offset: 1

Stefano Spezia, Feb 12 2022

			2.0838497097210232082242192894961701...

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Theorem 2, p. 2.

Cf. A082640.
Cf. A351480, A351481, A351482.
Cf. A351484, A351485, A351486, A351487, A351488.

cp's OEIS Frontend

A082640 Triangle T(m,n) read by rows: unimodular triangulations of the grid P(m,n), m,n > 0, n <= m.

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A351484 a(n) is the number of primitive triangulations of a 5 X n rectangle.

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A351485 a(n) is the number of primitive triangulations of a 6 X n rectangle.

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A351486 a(n) is the number of primitive triangulations of a 7 X n rectangle.

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A351487 a(n) is the number of primitive triangulations of an 8 X n rectangle.

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A351480 Decimal expansion of (611 + sqrt(73))/36.

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A351481 Decimal expansion of log_2((611 + sqrt(73))/36)/2.

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A351482 Decimal expansion of lim_{n->oo} f(3, n)^(1/(3*n)), where f(m, n) is the number of primitive lattice triangulations of m X n rectangle.

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A351483 Decimal expansion of log_2(lim_{n->oo} f(3, n)^(1/(3*n))), where f(m, n) is the number of primitive lattice triangulations of m X n rectangle.

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