A212801 -id:A212801 - cp's OEIS frontend

A006240 Row 4 of array in A212801.

1, 40, 793, 12800, 193721, 2886520, 42999713, 642355200, 9617422321, 144167168200, 2162192792233, 32433400563200, 486521516676521, 7298047169453080, 109472483776866353, 1642098503032012800, 24631532723767204321, 369473147671033293160, 5542096617629211606073, 83131435057615545920000

Offset: 1

N. J. A. Sloane

nonn

Number of Eulerian circuits in the Cartesian product of two directed cycles of lengths 4 and n. - Andrew Howroyd, Jan 14 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..200
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.
Eric Weisstein's World of Mathematics, Checkers.

Cf. A212801.

T[m_, n_] := Product[2 - Exp[2*I*h*Pi/m] - Exp[2*I*k*Pi/n], {h, 1, m - 1}, {k, 1, n - 1}];
a[n_] := T[4, n] // Round;
Array[a, 20] (* Jean-François Alcover, Jul 04 2018 *)

Empirical g.f.: x*(1-167*x^2+1200*x^3-2505*x^4+3375*x^6)/((1-x)*(1-3*x)*(1-5*x)*(1-15*x)*(1-4*x+5*x^2)*(1-12*x+45*x^2)). - Bruno Berselli, May 31 2012

Empirical closed form: a(n) = (15^n+3^n-5^n-1+(2+i)^n+(2-i)^n -(6+3*i)^n -(6-3*i)^n)/4, where i=sqrt(-1). - Bruno Berselli, May 31 2012

Revised by N. J. A. Sloane, May 27 2012

A006239 Row 3 of array in A212801.

Original entry on oeis.org

1, 13, 108, 793, 5611, 39312, 274933, 1923025, 13455396, 94169413, 659134543, 4613813568, 32296413241, 226074381637, 1582520088348, 11077641280225, 77543496352291, 542804506787088, 3799631657379853, 26597421924762793

Offset: 1

N. J. A. Sloane

nonn

Number of Eulerian circuits in the Cartesian product of two directed cycles of lengths 3 and n. - Andrew Howroyd, Jan 14 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..200
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 211.
Eric Weisstein's World of Mathematics, Checkers.

Cf. A212801.

T[m_, n_] := Product[2 - Exp[2*I*h*Pi/m] - Exp[2*I*k*Pi/n], {h, 1, m - 1}, {k, 1, n - 1}];
a[n_] := T[3, n] // Round;
Array[a, 20] (* Jean-François Alcover, Jul 04 2018 *)

Empirical g.f.: x*(1-7*x^2)/((1-x)*(1-7*x)*(1-5*x+7*x^2)). - Bruno Berselli, May 31 2012

Empirical closed form: a(n) = (2^n*(1+7^n) -(5-i*sqrt(3))^n -(5+i*sqrt(3))^n) / (3*2^n), where i=sqrt(-1). - Bruno Berselli, May 31 2012

Revised by N. J. A. Sloane, May 27 2012

A212802 Row 5 of array in A212801.

Original entry on oeis.org

1, 121, 5611, 193721, 6050000, 183990301, 5598183221, 171567260161, 5290933752571, 163756656800000, 5076226921767101, 157423577321804321, 4881873153941565211, 151371085451034210421, 4692977668021522550000, 145487069742178319930401

Offset: 1

N. J. A. Sloane, May 27 2012

nonn

Number of Eulerian circuits in the Cartesian product of two directed cycles of lengths 5 and n. - Andrew Howroyd, Jan 12 2018

Andrew Howroyd, Table of n, a(n) for n = 1..200
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.

Cf. A212801.

T[m_, n_] := Product[2 - Exp[2*I*h*Pi/m] - Exp[2*I*k*Pi/n], {h, 1, m - 1}, {k, 1, n - 1}];
a[n_] := T[5, n] // Round;
Array[a, 20] (* Jean-François Alcover, Jul 04 2018 *)

From Andrew Howroyd, Jan 12 2018: (Start)
Empirical: a(n) = 121*a(n-1) - 6520*a(n-2) + 209330*a(n-3) - 4493120*a(n-4) + 68446433*a(n-5) - 766303183*a(n-6) + 6438802040*a(n-7) - 41070618160*a(n-8) + 199602863240*a(n-9) - 736417358863*a(n-10) + 2039087685503*a(n-11) - 4149490675520*a(n-12) + 5992940178830*a(n-13) - 5786524000120*a(n-14) + 3329026307431*a(n-15) - 852891037441*a(n-16) for n > 16.
Empirical g.f.: x*(1 - 31*x^2)*(1 - 2479*x^2 + 94380*x^3 - 1719180*x^4 + 18597458*x^5 - 128373600*x^6 + 576521198*x^7 - 1652131980*x^8 + 2811674580*x^9 - 2289408559*x^10 + 887503681*x^12)/((1 - x)*(1 - 31*x)*(1 - 12*x + 31*x^2)*(1 - 9*x + 31*x^2 - 49*x^3 + 31*x^4)*(1 - 19*x + 151*x^2 - 589*x^3 + 961*x^4)*(1 - 49*x + 961*x^2 - 8649*x^3 + 29791*x^4)).
(End)

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

Original entry on oeis.org

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.

A298117 Array read by antidiagonals: T(m,n) = number of Eulerian cycles in the torus grid graph C_m X C_n.

Original entry on oeis.org

2, 4, 4, 8, 40, 8, 16, 320, 320, 16, 32, 2368, 8616, 2368, 32, 64, 16832, 207496, 207496, 16832, 64, 128, 116608, 4788808, 15639936, 4788808, 116608, 128, 256, 793088, 108326760, 1116199200, 1116199200, 108326760, 793088, 256

Offset: 1

Andrew Howroyd, Jan 12 2018

			Array begins:
=========================================================================
m\n|  1      2         3           4              5                 6
---|---------------------------------------------------------------------
1  |  2      4         8          16             32                64 ...
2  |  4     40       320        2368          16832            116608 ...
3  |  8    320      8616      207496        4788808         108326760 ...
4  | 16   2368    207496    15639936     1116199200       77643032832 ...
5  | 32  16832   4788808  1116199200   242230440480    50917973008584 ...
6  | 64 116608 108326760 77643032832 50917973008584 32098460087825856 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..105
Eric Weisstein's World of Mathematics, Eulerian Cycle
Eric Weisstein's World of Mathematics, Torus Grid Graph

Rows 2..4 are A298198, A282621, A298197.
Main diagonal is A297385.
Cf. A212801, A298119.

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

cp's OEIS Frontend

A006240 Row 4 of array in A212801.

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A006239 Row 3 of array in A212801.

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A212802 Row 5 of array in A212801.

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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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A298117 Array read by antidiagonals: T(m,n) = number of Eulerian cycles in the torus grid graph C_m X C_n.

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

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