A321172 -id:A321172 - cp's OEIS frontend

A003763 Number of (undirected) Hamiltonian cycles on a 2n X 2n square grid of points.

1, 6, 1072, 4638576, 467260456608, 1076226888605605706, 56126499620491437281263608, 65882516522625836326159786165530572, 1733926377888966183927790794055670829347983946, 1020460427390768793543026965678152831571073052662428097106

Offset: 1

Jeffrey Shallit, Feb 14 2002

Orientation of the path is not important; you can start going either clockwise or counterclockwise.
The number is zero for a 2n+1 X 2n+1 grid (but see A222200).
These are also called "closed rook tours".

			a(1) = 1 because there is only one such path visiting all nodes of a square.

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

N. J. A. Sloane, Table of n, a(n) for n = 1..13 (first 11 terms from _Artem M. Karavaev_, Sep 29 2010; a(12) and a(13) from Pettersson, 2014, added by _N. J. A. Sloane_, Jun 05 2015)
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
J. L. Jacobsen, Exact enumeration of Hamiltonian circuits, walks and chains in two and three dimensions, J. Phys. A: Math. Theor. 40 (2007) 14667-14678.
Artem M. Karavaev, Hamilton Cycles.
Ville H. Pettersson, Enumerating Hamiltonian Cycles, The Electronic Journal of Combinatorics, Volume 21, Issue 4, 2014.
Ville Pettersson, Graph Algorithms for Constructing and Enumerating Cycles and Related Structures, Dissertation, Aalto, Finland, 2015.
A. Pönitz, Computing invariants in graphs of small bandwidth, Mathematics in Computers and Simulation, 49(1999), 179-191.
A. Pönitz, Über eine Methode zur Konstruktion... PhD Thesis (2004) C.3.
T. G. Schmalz, G. E. Hite and D. J. Klein, Compact self-avoiding circuits on two-dimensional lattices, J. Phys. A 17 (1984), 445-453.
N. J. A. Sloane, Illustration of a(2) = 6.
Peter Tittmann, Enumeration in graphs: counting Hamiltonian cycles. [Broken link?]
Peter Tittmann, Enumeration in graphs: counting Hamiltonian cycles. [Backup copy of top page only, on the Internet Archive]
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv preprint arXiv:1402.0545 [math.CO], 2014.
Index entries for sequences related to graphs, Hamiltonian.

Other enumerations of Hamiltonian cycles on a square grid: A120443, A140519, A140521, A222200, A222201.

a(n) = A321172(2n,2n). - Robert FERREOL, Apr 01 2019

Two more terms from Andre Poenitz [André Pönitz] and Peter Tittmann (poenitz(AT)htwm.de), Mar 03 2003
a(8) from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 21 2006
a(9) and a(10) from Jesper L. Jacobsen (jesper.jacobsen(AT)u-psud.fr), Dec 12 2007

A332307 Array read by antidiagonals: T(m,n) is the number of (undirected) Hamiltonian paths in the m X n grid graph.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 14, 20, 14, 1, 1, 22, 62, 62, 22, 1, 1, 32, 132, 276, 132, 32, 1, 1, 44, 336, 1006, 1006, 336, 44, 1, 1, 58, 688, 3610, 4324, 3610, 688, 58, 1, 1, 74, 1578, 12010, 26996, 26996, 12010, 1578, 74, 1, 1, 92, 3190, 38984, 109722, 229348, 109722, 38984, 3190, 92, 1

Offset: 1

Andrew Howroyd, Feb 09 2020

			Array begins:
================================================
m\n | 1  2   3     4      5       6        7
----+-------------------------------------------
  1 | 1  1   1     1      1       1        1 ...
  2 | 1  4   8    14     22      32       44 ...
  3 | 1  8  20    62    132     336      688 ...
  4 | 1 14  62   276   1006    3610    12010 ...
  5 | 1 22 132  1006   4324   26996   109722 ...
  6 | 1 32 336  3610  26996  229348  1620034 ...
  7 | 1 44 688 12010 109722 1620034 13535280 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..435
J. L. Jacobsen, Exact enumeration of Hamiltonian circuits, walks and chains in two and three dimensions, J. Phys. A: Math. Theor. 40 (2007) 14667-14678.
J.-M. Mayer, C. Guez and J. Dayantis, Exact computer enumeration of the number of Hamiltonian paths in small square plane lattices, Physical Review B, Vol. 42 Number 1, 1990.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path

Rows n=1..9 are A000012, A003682, A003685, A003695, A003778, A145402, A358794, A358795, A358796.
Main diagonal is A120443.
Cf. A064298, A231829, A271465, A271592, A288518, A321172.

T(n,m) = T(m,n).

A339190 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of (undirected) Hamiltonian cycles on the n X k king graph.

Original entry on oeis.org

3, 4, 4, 8, 16, 8, 16, 120, 120, 16, 32, 744, 2830, 744, 32, 64, 4922, 50354, 50354, 4922, 64, 128, 31904, 1003218, 2462064, 1003218, 31904, 128, 256, 208118, 19380610, 139472532, 139472532, 19380610, 208118, 256, 512, 1354872, 378005474, 7621612496, 22853860116, 7621612496, 378005474, 1354872, 512

Offset: 2

Seiichi Manyama, Nov 27 2020

			Square array T(n,k) begins:
   3,     4,        8,         16,            32,               64, ...
   4,    16,      120,        744,          4922,            31904, ...
   8,   120,     2830,      50354,       1003218,         19380610, ...
  16,   744,    50354,    2462064,     139472532,       7621612496, ...
  32,  4922,  1003218,  139472532,   22853860116,    3601249330324, ...
  64, 31904, 19380610, 7621612496, 3601249330324, 1622043117414624, ...

Seiichi Manyama, Antidiagonals n = 2..12, flattened
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, King Graph
Index entries for sequences related to graphs, Hamiltonian

Rows and columns 3..5 give A339200, A339201, A339202.
Main diagonal gives A140519.
Cf. A321172, A339098, A339849.

# Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
            if i < k:
                grids.append((i + (j - 1) * k, i + j * k + 1))
            if i > 1:
                grids.append((i + (j - 1) * k, i + j * k - 1))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
    return grids
def A339190(n, k):
    universe = make_nXk_king_graph(n, k)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles(is_hamilton=True)
    return cycles.len()
print([A339190(j + 2, i - j + 2) for i in range(10 - 1) for j in range(i + 1)])

T(n,k) = T(k,n).

A359855 Array read by antidiagonals: T(n,k) is the number of Hamiltonian cycles in the stacked prism graph P_n X C_k, n >= 1, k >= 2.

Original entry on oeis.org

1, 1, 4, 1, 3, 4, 1, 6, 6, 4, 1, 5, 22, 12, 4, 1, 8, 30, 82, 24, 4, 1, 7, 86, 160, 306, 48, 4, 1, 10, 126, 776, 850, 1142, 96, 4, 1, 9, 318, 1484, 7010, 4520, 4262, 192, 4, 1, 12, 510, 6114, 18452, 63674, 24040, 15906, 384, 4, 1, 11, 1182, 12348, 126426, 229698, 578090, 127860, 59362, 768, 4

Offset: 1

Andrew Howroyd, Feb 18 2025

The case for P_n X C_2 is determined using a double edge for C_2.

			Array begins:
=========================================================
n\k | 2   3     4      5       6        7          8 ...
----+---------------------------------------------------
  1 | 1   1     1      1       1        1          1 ...
  2 | 4   3     6      5       8        7         10 ...
  3 | 4   6    22     30      86      126        318 ...
  4 | 4  12    82    160     776     1484       6114 ...
  5 | 4  24   306    850    7010    18452     126426 ...
  6 | 4  48  1142   4520   63674   229698    2588218 ...
  7 | 4  96  4262  24040  578090  2861964   53055038 ...
  8 | 4 192 15906 127860 5247824 35663964 1087362018 ...
   ...

Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Stacked Prism Graph.

Columns 2..12 are A123932(n-1), A003945(n-1), A003699, A003731, A180582, A180583, A180584, A180585, A180586, A180587, A180588.
Rows 1..2 are A000012, A103889(n+1).
Cf. A222196 (order of recurrences), A222197 (main diagonal), A270273, A321172.

A006864 Number of Hamiltonian cycles in P_4 X P_n.

Original entry on oeis.org

0, 1, 2, 6, 14, 37, 92, 236, 596, 1517, 3846, 9770, 24794, 62953, 159800, 405688, 1029864, 2614457, 6637066, 16849006, 42773094, 108584525, 275654292, 699780452, 1776473532, 4509783909, 11448608270, 29063617746, 73781357746, 187302518353, 475489124976, 1207084188912

Offset: 1

kwong(AT)cs.fredonia.edu (Harris Kwong), N. J. A. Sloane, Simon Plouffe and Frans J. Faase

Wazir tours on a 4 X n grid. There are two closed loops for a 4x4 board, appearing as an H and a C, for example. - Ed Pegg Jr, Sep 07 2010

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
Kwong, Y. H. H.; Enumeration of Hamiltonian cycles in P_4 X P_n and P_5 X P_n. Ars Combin. 33 (1992), 87-96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Huaide Cheng, Table of n, a(n) for n = 1..2473
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
C. Flye Sainte-Marie, Manières différentes de tracer une route fermée ..., L'Intermédiaire des Mathématiciens, vol. 11 (1904), pp. 86-88 (in French).
George Jelliss, Wazir Wanderings
R. Tosic, O. Bodroza, Y. H. Harris Kwong, and H. Joseph Straight, On the number of Hamiltonian cycles of P4 X Pn, Indian J. Pure Appl. Math. 21 (5) (1990), 403-409.
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,1).

Row/column 4 of A321172.

LinearRecurrence[{2, 2, -2, 1}, {0, 1, 2, 6}, 50] (* Paolo Xausa, Jul 01 2025 *)

a(n):=sum ( sum ( binomial(k,j) *sum (binomial(j, i-j)*2^j *binomial(k-j,n-i-3*(k-j))*(-2)^(4*(k-j)-(n-i)), i,j,n-k+j) , j,0,k) , k,1,n ); /* Vladimir Kruchinin, Aug 04 2010 */

a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) + a(n-4).
G.f.: x^2/(1-2x-2x^2+2x^3-x^4). - R. J. Mathar, Dec 16 2008
a(n)=sum ( sum ( binomial(k,j) * sum (binomial(j, i-j)*2^j *binomial(k-j,n-i-3*(k-j))*(-2)^(4*(k-j)-(n-i)), i,j,n-k+j) , j,0,k) , k,1,n ), n>0. - Vladimir Kruchinin, Aug 04 2010
a(n) = Sum_{k=1..n-1} A181688(k). - Kevin McShane, Aug 04 2019

A145418 Number of Hamiltonian cycles in P_8 X P_n.

Original entry on oeis.org

0, 1, 8, 236, 1696, 32675, 301384, 4638576, 49483138, 681728204, 7837276902, 102283239429, 1220732524976, 15513067188008, 188620289493918, 2365714170297014, 29030309635705054, 361749878496079778, 4459396682866920534, 55391169255983979555, 684363209103066303906

Offset: 1

N. J. A. Sloane, Feb 03 2009

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Alois P. Heinz, Table of n, a(n) for n = 1..917
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs.
F. Faase, Results from the counting program
A. Kloczkowski, and R. L. Jernigan, Transfer matrix method for enumeration and generation of compact self-avoiding walks. I. Square lattices, The Journal of Chemical Physics 109, 5134 (1998); doi: 10.1063/1.477128
Index entries for linear recurrences with constant coefficients, order 66.

Row/column 8 of A321172.

Recurrence:
a(1) = 0,
a(2) = 1,
a(3) = 8,
a(4) = 236,
a(5) = 1696,
a(6) = 32675,
a(7) = 301384,
a(8) = 4638576,
a(9) = 49483138,
a(10) = 681728204,
a(11) = 7837276902,
a(12) = 102283239429,
a(13) = 1220732524976,
a(14) = 15513067188008,
a(15) = 188620289493918,
a(16) = 2365714170297014,
a(17) = 29030309635705054,
a(18) = 361749878496079778,
a(19) = 4459396682866920534,
a(20) = 55391169255983979555,
a(21) = 684363209103066303906,
a(22) = 8487168277379774266411,
a(23) = 104976660007043902770814,
a(24) = 1300854247070195164448395,
a(25) = 16098959403506801921858124,
a(26) = 199418506963731877069653608,
a(27) = 2468612432237087475265791106,
a(28) = 30572953033472980838613625389,
a(29) = 378515201134457658578140498814,
a(30) = 4687342384540802154353083423651,
a(31) = 58036542374043013796287237537528,
a(32) = 718661780960820074611282900026324,
a(33) = 8898436384928204979882033571220340,
a(34) = 110186062841343288284017151289070451,
a(35) = 1364340857418682291195543074012508456,
a(36) = 16893937354451697990213722467612836695,
a(37) = 209185026496655279949634983839901418774,
a(38) = 2590216891342324056714821054881440813215,
a(39) = 32072851564440568180804318145788811014976,
a(40) = 397138412927090582354377476417693090903768,
a(41) = 4917498017559613255667946000320694921175130,
a(42) = 60890272030773519479287882832089863209466478,
a(43) = 753964042571110322417001735829736156594209380,
a(44) = 9335854145287983656933756936219959893935498622,
a(45) = 115599774527478742012501648761874199775452411672,
a(46) = 1431397531309770867365502551162804883408923187965,
a(47) = 17724063449625564471462425816551511960390740556400,
a(48) = 219465622040057380709984287099015972930644329156424,
a(49) = 2717500192865830096645192106030659520142409708395450,
a(50) = 33649045694807090450997457881543310615794538874090382,
a(51) = 416654292509213357722564031894407450765035835407734706,
a(52) = 5159160169073567278327353311624938215272772058329334389,
a(53) = 63882533593051394161814876759814129552293422016852019728,
a(54) = 791016010339998093452532578418540484158488096782539430286,
a(55) = 9794638258031421885388598947932945990242328205117007130718,
a(56) = 121280656298395438005330895082043790844069204530565536980402,
a(57) = 1501739723290424387359817153191514221861132297169144591119746,
a(58) = 18595069417782079319375695239542203044044419158097555496277590,
a(59) = 230250687548524273220393339819664989761608497977237213691651494,
a(60) = 2851044985755900792432116853155397844049903269953868448269465911,
a(61) = 35302641500328319561839557836179860373923985349499838565583491438,
a(62) = 437129721450539018107540085474755888131298517879956664876467411931,
a(63) = 5412693919496858591306748921846182243342130551030595689565457284562,
a(64) = 67021879478670244241238920776850020175011969240135534404057401625317,
a(65) = 829888479044613035646707314461069153586129302554576136417149736843676,
a(66) = 10275970973805259625689798376883875013812168498330812425399678612679778, and
a(n) = 16a(n-1) + 59a(n-2) - 1824a(n-3) + 3898a(n-4) + 55218a(n-5)
- 243282a(n-6) - 545916a(n-7) + 4861689a(n-8) - 2576498a(n-9) - 43488068a(n-10)
+ 94333210a(n-11) + 141446298a(n-12) - 752431432a(n-13) + 377840445a(n-14) + 2789611474a(n-15)
- 4656548198a(n-16) - 5258354388a(n-17) + 18170944298a(n-18) + 3512822542a(n-19) - 45026326037a(n-20)
+ 9980240588a(n-21) + 84208620015a(n-22) - 44876200668a(n-23) - 121497215791a(n-24) + 102246696772a(n-25)
+ 117755621290a(n-26) - 145213823124a(n-27) - 60571088405a(n-28) + 136877858022a(n-29) + 3649170978a(n-30)
- 100110796416a(n-31) + 42689760462a(n-32) + 39482359310a(n-33) - 72614614806a(n-34) + 27495494908a(n-35)
+ 40732692257a(n-36) - 38863698070a(n-37) + 9092063794a(n-38) + 5076214026a(n-39) - 9600155591a(n-40)
+ 4294619636a(n-41) - 1463899423a(n-42) + 4331661320a(n-43) - 2669382577a(n-44) - 998576578a(n-45)
+ 1722204514a(n-46) - 1646502104a(n-47) + 1188567443a(n-48) - 143652474a(n-49) - 380794039a(n-50)
- 27735814a(n-51) + 132682964a(n-52) + 79877148a(n-53) + 41238077a(n-54) - 16408310a(n-55)
- 42867025a(n-56) - 18129698a(n-57) + 4261277a(n-58) + 4951334a(n-59) + 985598a(n-60)
- 103168a(n-61) - 13629a(n-62) + 34282a(n-63) + 6952a(n-64) - 532a(n-65)
+ 36a(n-66).

A145401 Number of Hamiltonian cycles in P_6 X P_n.

Original entry on oeis.org

0, 1, 4, 37, 154, 1072, 5320, 32675, 175294, 1024028, 5668692, 32463802, 181971848, 1033917350, 5824476298, 32989068162, 186210666468, 1053349394128, 5950467515104, 33643541208290, 190115484271760, 1074685815276400, 6073680777522430, 34330607094625734

Offset: 1

N. J. A. Sloane, Feb 03 2009

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Huaide Cheng, Table of n, a(n) for n = 1..1331 (terms 1..400 from Alois P. Heinz)
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs.
F. Faase, Results from the counting program
Index entries for linear recurrences with constant coefficients, signature (5,14,-63,12,90,-35,-66,118,-8,-82,42,28,-4,2).

Row/column 6 of A321172.

a:= n-> (Matrix([32675, 5320, 1072, 154, 37, 4, 1, 0, 1/2, 0, -5, -16, 51, 869/2]). Matrix(14, (i, j)-> if i=j-1 then 1 elif j=1 then [5, 14, -63, 12, 90, -35, -66, 118, -8, -82, 42, 28, -4, 2][i] else 0 fi)^n)[1,9]: seq(a(n), n=1..30);  # Alois P. Heinz, Oct 24 2009

a[n_] := ({32675, 5320, 1072, 154, 37, 4, 1, 0, 1/2, 0, -5, -16, 51, 869/2 }.MatrixPower[Table[If[i == j-1, 1, If[j == 1, {5, 14, -63, 12, 90, -35, -66, 118, -8, -82, 42, 28, -4, 2}[[i]], 0]], {i, 1, 14}, {j, 1, 14}], n] )[[9]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 14 2016, after Alois P. Heinz *)

Recurrence: a(1) = 0,
a(2) = 1,
a(3) = 4,
a(4) = 37,
a(5) = 154,
a(6) = 1072,
a(7) = 5320,
a(8) = 32675,
a(9) = 175294,
a(10) = 1024028,
a(11) = 5668692,
a(12) = 32463802,
a(13) = 181971848,
a(14) = 1033917350, and
a(n) = 5a(n-1) +14a(n-2) -63a(n-3) +12a(n-4) +90a(n-5) -35a(n-6) -66a(n-7) +118a(n-8) -8a(n-9) -82a(n-10) +42a(n-11) +28a(n-12) -4a(n-13) +2a(n-14).
G.f.: -x^2*(x -1)*(x^11 -x^10 +3*x^9 +12*x^8 -3*x^7 -3*x^4 +21*x^3 -3*x^2 -1)/(2*x^14 -4*x^13 +28*x^12 +42*x^11 -82*x^10 -8*x^9 +118*x^8 -66*x^7- 35*x^6 +90*x^5 +12*x^4 -63*x^3 +14*x^2 +5*x -1). [Colin Barker, Aug 31 2012]

More terms from Alois P. Heinz, Oct 24 2009

A145416 Number of Hamiltonian cycles in P_7 X P_2n.

Original entry on oeis.org

1, 92, 5320, 301384, 17066492, 966656134, 54756073582, 3101696069920, 175698206778318, 9952578156814524, 563772503196695338, 31935387285412942410, 1809007988782552388490, 102472842263117124008066, 5804663918990466729365476, 328810272735298761062754308

Offset: 1

N. J. A. Sloane, Feb 03 2009

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Huaide Cheng, Table of n, a(n) for n = 1..571 (terms 1..500 from Seiichi Manyama)
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs.
F. Faase, Results from the counting program
Index entries for linear recurrences with constant coefficients, signature (85,-1932,20403,-116734,386724,-815141,1251439,-1690670,2681994,-4008954,3390877,-1036420,-178842,92790,17732,-5972,1728,144).

Cf. A321172.

Recurrence:
a(1) = 1,
a(2) = 92,
a(3) = 5320,
a(4) = 301384,
a(5) = 17066492,
a(6) = 966656134,
a(7) = 54756073582,
a(8) = 3101696069920,
a(9) = 175698206778318,
a(10) = 9952578156814524,
a(11) = 563772503196695338,
a(12) = 31935387285412942410,
a(13) = 1809007988782552388490,
a(14) = 102472842263117124008066,
a(15) = 5804663918990466729365476,
a(16) = 328810272735298761062754308,
a(17) = 18625745945872429428768223714,
a(18) = 1055071695766249759732087999456, and
a(n) = 85a(n-1) - 1932a(n-2) + 20403a(n-3) - 116734a(n-4) + 386724a(n-5)
- 815141a(n-6) + 1251439a(n-7) - 1690670a(n-8) + 2681994a(n-9)
- 4008954a(n-10) + 3390877a(n-11) - 1036420a(n-12) - 178842a(n-13)
+ 92790a(n-14) + 17732a(n-15) - 5972a(n-16) + 1728a(n-17) + 144a(n-18).

Recurrence corrected by Frans J. Faase, Feb 04 2009

A180504 Number of Hamiltonian cycles in P_10 X P_n.

Original entry on oeis.org

0, 1, 16, 1517, 18684, 1024028, 17066492, 681728204, 13916993782, 467260456608, 10754797724124, 328076475659033, 8091313110371792, 233977398720987284, 6002042996016384360, 168435972906750526954, 4418118886987754341770, 121913396076344218930045

Offset: 1

Artem M. Karavaev, Sep 09 2010

nonn

The linear recurrence for this sequence has order 346. It is too large to be posted here.

Huaide Cheng, Table of n, a(n) for n = 1..700 (terms 1..100 from Andrew Howroyd)
Flow Problem, Hamilton Cycles (see Maple worksheet).
A. Kloczkowski and R. L. Jernigan, Transfer matrix method for enumeration and generation of compact self-avoiding walks. I. Square lattices, J. Chem. Phys., 109 (1998), 5134-5146. See Table IV, column m = 10.
Index entries for sequences related to graphs, Hamiltonian
Index entries for linear recurrences with constant coefficients, order 346.

Row/column 10 of A321172.

Cf. A160149.

a(16) onwards from Andrew Howroyd, Dec 13 2024

A180505 Number of Hamiltonian cycles in P_11 X P_2n.

Original entry on oeis.org

1, 3846, 5668692, 7837276902, 10754797724124, 14746957510647992, 20223692320200140940, 27738606105535271640888, 38049128385426605236700966, 52194036750499722755908743018, 71598455565101470929617326988084, 98217523834843365306426848969040826, 134733398926676359394934062807293332148

Offset: 1

Artem M. Karavaev, Sep 09 2010

nonn

The linear recurrence for this sequence has order 671. It is too large to be posted here.

Huaide Cheng, Table of n, a(n) for n = 1..319 (terms 1..50 Seiichi Manyama)
Flow Problem, Hamilton Cycles (see Maple worksheet).
Index entries for linear recurrences with constant coefficients, order 671.

Cf. A180504, A321172.

a(11)-a(13) from Seiichi Manyama, Mar 29 2020

cp's OEIS Frontend

A003763 Number of (undirected) Hamiltonian cycles on a 2n X 2n square grid of points.

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A332307 Array read by antidiagonals: T(m,n) is the number of (undirected) Hamiltonian paths in the m X n grid graph.

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A339190 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of (undirected) Hamiltonian cycles on the n X k king graph.

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A359855 Array read by antidiagonals: T(n,k) is the number of Hamiltonian cycles in the stacked prism graph P_n X C_k, n >= 1, k >= 2.

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A006864 Number of Hamiltonian cycles in P_4 X P_n.

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A145418 Number of Hamiltonian cycles in P_8 X P_n.

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A145401 Number of Hamiltonian cycles in P_6 X P_n.

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A145416 Number of Hamiltonian cycles in P_7 X P_2n.

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A180504 Number of Hamiltonian cycles in P_10 X P_n.