A054759 -id:A054759 - cp's OEIS frontend

A298119 Array read by antidiagonals: T(m,n) = number of Eulerian orientations of the torus grid graph C_m X C_n.

4, 8, 8, 16, 18, 16, 32, 44, 44, 32, 64, 114, 148, 114, 64, 128, 308, 548, 548, 308, 128, 256, 858, 2116, 2970, 2116, 858, 256, 512, 2444, 8324, 16892, 16892, 8324, 2444, 512, 1024, 7074, 33028, 98466, 143224, 98466, 33028, 7074

Offset: 1

Andrew Howroyd, Jan 12 2018

In other words, the number of orientations of the m X n torus grid graph in which each vertex has equal indegree and outdegree.

Values are always even since reversing the orientation of each edge will always result in another Eulerian orientation.

			Array begins:
============================================================
m\n|   1    2     3      4        5         6          7
---|--------------------------------------------------------
1  |   4    8    16     32       64       128        256 ...
2  |   8   18    44    114      308       858       2444 ...
3  |  16   44   148    548     2116      8324      33028 ...
4  |  32  114   548   2970    16892     98466     583412 ...
5  |  64  308  2116  16892   143224   1250228   11091536 ...
6  | 128  858  8324  98466  1250228  16448400  220603364 ...
7  | 256 2444 33028 583412 11091536 220603364 4484823396 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..91
Eric Weisstein's World of Mathematics, Torus Grid Graph

Main diagonal is A054759.
Rows 2..5 are 2*A099754, 2*A170938, A298201, A372093, A372094.
Cf. A212801, A298117.

A118273 Decimal expansion of (4/3)^(3/2).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 21 2006

The volume of the cube inscribed in the unit-radius sphere. - Amiram Eldar, Jun 02 2023

			1.539600717839002038...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.24, p. 412.

Elliott H. Lieb, Residual entropy of square ice, Phys. Rev. 162 (1967) 162-172.
James Propp, The many faces of alternating-sign matrices, 2002.
Scenta server, Lieb's square ice constant. [Broken link]
Eric Weisstein's World of Mathematics, Lieb's Square Ice Constant.
Wikipedia, Lieb's square ice constant.
Index entries for algebraic numbers, degree 2.

Cf. A020784, A054759.

Cf. A122553 (octahedron), A339259 (regular icosahedron), A363437 (regular tetrahedron), A363438 (regular dodecahedron).

evalf(sqrt((4/3)^3)) ; # R. J. Mathar, Mar 29 2012

RealDigits[(4/3)^(3/2),10,120][[1]] (* Harvey P. Dale, Aug 06 2015 *)

(4/3)^(3/2) \\ Stefano Spezia, Dec 15 2024

Equals 8 * A020784.

A212803 Number of Eulerian circuits in the Cartesian product of two directed cycles of lengths n.

Original entry on oeis.org

1, 4, 108, 12800, 6050000, 11218701312, 81959473720768, 2376692369090150400, 275204089028043534645504, 127722545775271195902771200000, 238045190395699755964859156456705024, 1783083199654005767436422099232872202240000, 53684915729010675246823790713834564866472376291328

Offset: 1

N. J. A. Sloane, May 27 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.

Main diagonal of A212801.

Cf. A054759, A297385.

a[n_] := Product[2-E^(2 h Pi I/n)-E^(2 k Pi I/n), {h, 1, n-1}, {k, 1, n-1}];
Array[a, 12] // Round (* Jean-François Alcover, Sep 02 2019 *)

Name clarified by Andrew Howroyd, Jan 12 2018

Terms a(13) and beyond from Andrew Howroyd, May 19 2020

A297385 Number of Eulerian cycles in the n X n torus grid graph.

Original entry on oeis.org

2, 40, 8616, 15639936, 242230440480, 32098460087825856, 36433115190009846104160

Offset: 1

Eric W. Weisstein, Dec 29 2017

a(3) is also the number of Eulerian cycles in the 3 X 3 rook graph.

a(4) is also the number of Eulerian cycles in the 4-hypercube (tesseract) graph Q_4.

Eric Weisstein's World of Mathematics, Eulerian Cycle
Eric Weisstein's World of Mathematics, Torus Grid Graph

Main diagonal of A298117.

Cf. A054759, A212803.

a(4)-a(7) from Andrew Howroyd, Jan 12 2018

a(1)-a(2) from Andrew Howroyd, Jan 12 2018

A358177 Number of Eulerian orientations of a (labeled) 2n-dimensional hypercube graph, Q_2n. Q_2n is also the n-dimensional torus grid graph (C_4)^n.

Original entry on oeis.org

1, 2, 2970, 351135773356461511142023680

Offset: 0

Peter Munn and Zachary DeStefano, Nov 02 2022

An Eulerian orientation of a graph is an orientation of the edges such that every vertex has in-degree equal to out-degree. (C_4)^n denotes the Cartesian product of n cycle graphs on 4 nodes.

			For n = 1, dimension 2n = 2, there are two Eulerian orientations (the cyclic ones). So a(1) = 2.

Mikhail Isaev, Brendan D. McKay, and Rui-Ray Zhang, Correlation between residual entropy and spanning tree entropy of ice-type models on graphs, arXiv:2409.04989 [math.CO], 2024. See p. 24.
A. Schrijver, Bounds on the Number of Eulerian Orientations, Combinatorica, 3 (3--4), (1983), 375-380.
Eric Weisstein's World of Mathematics, Cartesian product of graphs
Eric Weisstein's World of Mathematics, Hypercube Graph

Cf. A007081, A054759, A298119, A307334, A334553.

a(0) = A007081(2^0) = 1.
a(1) = A334553(1) = 2.
a(2) = A054759(4) = 2970.
Schrijver (1983) provides general bounds on unknown terms of the form (2^(-k) * binomial(2k,k))^(2^(2k)) <= a(k) <= sqrt(binomial(2k,k)^(2^(2k))).
From this we have the specific bounds 2.9*10^25 <= a(3) <= 4.3*10^41 and 1.2*10^164 <= a(4) <= 1.5*10^236.

a(3) added by Brendan McKay, Nov 04 2022

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A298119 Array read by antidiagonals: T(m,n) = number of Eulerian orientations of the torus grid graph C_m X C_n.

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A118273 Decimal expansion of (4/3)^(3/2).

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A212803 Number of Eulerian circuits in the Cartesian product of two directed cycles of lengths n.

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A297385 Number of Eulerian cycles in the n X n torus grid graph.

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A358177 Number of Eulerian orientations of a (labeled) 2n-dimensional hypercube graph, Q_2n. Q_2n is also the n-dimensional torus grid graph (C_4)^n.

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