A122722 -id:A122722 - cp's OEIS frontend

A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n(n + 1)/2, read by rows.

1, 1, 3, 1, 9, 24, 18, 1, 18, 126, 434, 762, 630, 187, 1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135, 1, 45, 915, 11127, 90270, 515970, 2139120, 6523428, 14683401, 24256853, 28975770, 24383838, 13860321, 4966929, 989970, 81462, 1, 63

Offset: 1

R. J. Mathar, May 23 2016

			Triangle T(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:
  1;
  1,  3;
  1,  9,  24,   18;
  1, 18, 126,  434,   762,   630,   187;
  1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135;
  ...

R. J. Mathar, Table of n, a(n) for n = 1..575
J. A. De Loera, J. Rambau, F. Santos, Further topics, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), 433-511.
R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004.
Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.
Wikipedia, Lozenge.

Cf. A045943 (column k=1), A011555, A011556, A011781, A122722, A326367 (k=2), A326368 (k=3), A326369 (k=4), A000124 (row lengths).

T(n,2) = 3*(n-1)*(n-2)*(3*n^2+3*n-4)/8 . - R. J. Mathar, May 24 2016
T(n,3) = (n-2)*(9*n^5-9*n^4-81*n^3+81*n^2+160*n-192)/16. - Greg Dresden, Jul 03 2019
Conjecture: T(n,4) = 3*(n-2)*(n-3)*(9*n^6+9*n^5-135*n^4-81*n^3+670*n^2+104*n-1216)/128. - Greg Dresden, Jul 03 2019
Conjecture: T(n,5) = 3*(n-3)*(n+3)* (27*n^8 -135*n^7 -387*n^6 +2835*n^5 -168*n^4 -18732*n^3 +19568*n^2 +36992*n -56320)/1280. - R. J. Mathar, Jul 07 2019
From Petros Hadjicostas, Sep 13 2019: (Start)
Conjecture for rightmost terms: A122722(n) =  n! * T(n, n*(n+1)/2) for n >= 1.
Conjectures for column k >= 0: Sum_{0 <= s <= 2*k + 1} (-1)^s * binomial(2*k+1, s) * T(n-s, k) = 0 for n >= 2*k+2.
Sum_{0 <= s <= 2*k} (-1)^s * binomial(2*k, s) * T(n-s, k) = A011781(k) for n >= 2*k+1. (End)

A244996 Decimal expansion of the moment derivative W_3'(0) associated with the radial probability distribution of a 3-step uniform random walk.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 09 2014

This constant is also associated with the asymptotic number of lozenge tilings; see the references by Santos (2004, 2005). It is called the "maximum asymptotic normalized entropy of lozenge tilings of a planar region". Santos (2004, 2005) mentions that is computed in Cohn et al. (2000). For discussion of lozenge tilings, see for example the references for sequences A122722 and A273464. - Petros Hadjicostas, Sep 13 2019

			0.3230659472194505140936365107238063940722418407805870161308684703610151128...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003; see Section 3.10, Kneser-Mahler Polynomial Constants, p. 232.

Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks, Canad. J. Math. 64(1) (2012), 961-990; see p. 978.
Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, arXiv:math/0008220 [math.CO], 2000.
Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14(2) (2000), 297-346.
Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004; see part (2) of Theorem 1 (p. 2, possible typo), Lemma 4.8 (p. 22), and Theorem 4.9 (p. 22).
Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.
Eric Weisstein's MathWorld, Clausen's Integral.
Eric Weisstein's MathWorld, Lobachevsky's Function.
Wikipedia, Lozenge.
Wikipedia, Clausen function.

Cf. A122722, A273464, A242710.

Clausen2[x_] := Im[PolyLog[2, Exp[x*I]]]; RealDigits[(1/Pi)*Clausen2[Pi/3], 10, 105] // First

imag(polylog(2,exp(Pi*I/3)))/Pi \\ Charles R Greathouse IV, Aug 27 2014

W_3'(0) = (1/Pi)*Cl2[Pi/3] = (3/(2*Pi))*Cl2[2*Pi/3], where Cl2 is the Clausen function.
W_3'(0) = integral_{y=1/6..5/6} log(2*sin(Pi*y)).
Also equals log(A242710).

A242710 Decimal expansion of "beta", a Kneser-Mahler polynomial constant (a constant related to the asymptotic evaluation of the supremum norm of polynomials).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 21 2014

			1.38135644451849779337146695685...

Steven R. Finch, Mathematical Constants, Cambridge, 2003; see Section 3.10, Kneser-Mahler polynomial constants, p. 232, and Section 5.23, Monomer-dimer constants, p. 408.

Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks, Canad. J. Math. 64(1) (2012), 961-990; see p. 978.
Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, arXiv:math/0008220 [math.CO], 2000.
Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14(2) (2000), 297-346.
Kurt Mahler, A remark on a paper of mine on polynomials. [In this paper, j is log(beta) = A244996.]
Kurt Mahler, A remark on a paper of mine on polynomials, Illinois J. Math. 8(1) (1964), 1-4.
Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004; see part (2) of Theorem 1 (p. 2, possible typo), Lemma 4.8 (p. 22), and Theorem 4.9 (p. 22).
Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.
Eric Weisstein's MathWorld, Clausen's Integral.
Eric Weisstein's MathWorld, Gieseking's Constant.
Eric Weisstein's MathWorld, Lobachevsky's Function.
Wikipedia, Clausen function.

Cf. A122722, A130834, A143298, A229728, A244996.

Exp[(PolyGamma[1, 4/3] - PolyGamma[1, 2/3] + 9)/(4*Sqrt[3]*Pi)] // RealDigits[#, 10, 100]& // First

beta = exp(G/Pi) = exp((PolyGamma(1, 4/3) - PolyGamma(1, 2/3) + 9)/(4*sqrt(3)*Pi)), where G is Gieseking's constant (cf. A143298) and PolyGamma(1,z) the first derivative of the digamma function psi(z).

Also equals exp(-Im(Li_2( 1/2 - (i*sqrt(3))/2))/Pi), where Li_2 is the dilogarithm function.

A011555 Number of vertices of secondary polytope for triangle X n-simplex.

Original entry on oeis.org

6, 108, 4488, 376200

Offset: 1

N. J. A. Sloane

Gelfand, Kapranov and Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1994, pp. 243-250.

J. A. de Loera, Nonregular triangulations of products of simplices, Discrete Comp. Geom. 15(3) (1996), 253-264.

Cf. A011556, A011781, A122722, A273464, A326367, A326368, A326369.

A011556 Number of regular triangulations of triangle X n-simplex.

Original entry on oeis.org

1, 5, 35, 530

Offset: 1

N. J. A. Sloane

Gelfand, Kapranov and Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1994, pp. 243-250.

J. A. de Loera, Nonregular triangulations of products of simplices, Discrete Comp. Geom. 15(3) (1996), 253-264.

Cf. A011555, A011781, A122722, A273464, A326367, A326368, A326369.

cp's OEIS Frontend

A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2*k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n*(n + 1)/2, read by rows.

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A244996 Decimal expansion of the moment derivative W_3'(0) associated with the radial probability distribution of a 3-step uniform random walk.

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A242710 Decimal expansion of "beta", a Kneser-Mahler polynomial constant (a constant related to the asymptotic evaluation of the supremum norm of polynomials).

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A011555 Number of vertices of secondary polytope for triangle X n-simplex.

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A011556 Number of regular triangulations of triangle X n-simplex.

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A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n(n + 1)/2, read by rows.