A332307 -id:A332307 - cp's OEIS frontend

A321172 Triangle read by rows: T(m,n) = number of Hamiltonian cycles on m X n grid of points (m >= 2, 2 <= n <= m).

1, 1, 0, 1, 2, 6, 1, 0, 14, 0, 1, 4, 37, 154, 1072, 1, 0, 92, 0, 5320, 0, 1, 8, 236, 1696, 32675, 301384, 4638576, 1, 0, 596, 0, 175294, 0, 49483138, 0, 1, 16, 1517, 18684, 1024028, 17066492, 681728204, 13916993782, 467260456608

Offset: 2

Robert FERREOL, Jan 10 2019

Orientation of the path is not important; you can start going either clockwise or counterclockwise. Paths related by symmetries are considered distinct.
The m X n grid of points when drawn forms a (m-1) X (n-1) rectangle of cells, so m-1 and n-1 are sometimes used as indices instead of m and n (see, e. g., Kreweras' reference below).
These cycles are also called "closed non-intersecting rook's tours" on m X n chess board.

			T(5,4)=14 is illustrated in the links above.
Table starts:
=================================================================
m\n|  2    3      4       5         6           7            8
---|-------------------------------------------------------------
2  |  1    1      1       1         1           1            1
3  |  1    0      2       0         4           0            8
4  |  1    2      6      14        37          92          236
5  |  1    0     14       0       154           0         1696
6  |  1    4     37     154      1072        5320        32675
7  |  1    0     92       0      5320           0       301384
8  |  1    8    236    1696     32675      301384      4638576
The table is symmetric, so it is completely described by its lower-left half.

Huaide Cheng, Rows n=2..17 of triangle, flattened
Robert Ferréol, The T(4,5)=14 hamiltonian cycles on 4 X 5 square grid of points; the T(5,6) = 154 cycles on 5 X 6 grid are reduced to 44 different forms.
Germain Kreweras, Dénombrement des cycles hamiltoniens dans un rectangle quadrillé, European Journal of Combinatorics, Volume 13, Issue 6 (1992), page 476.
Robert Stoyan and Volker Strehl, Enumeration of Hamiltonian Circuits in Rectangular Grids, Séminaire Lotharingien de Combinatoire, B34f (1995), 21 pp.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.

Row/column k=4..12 are: (with interspersed zeros for odd k): A006864, A006865, A145401, A145416, A145418, A160149, A180504, A180505, A213813.
Cf. A003763 (bisection of main diagonal), A222200 (subdiagonal), A231829, A270273, A332307.
T(n,2n) gives A333864.

# Program due to Laurent Jouhet-Reverdy
def cycle(m, n):
     if (m%2==1 and n%2==1): return 0
     grid = [[0]*n for _ in range(m)]
     grid[0][0] = 1; grid[1][0] = 1
     counter = [0]; stop = m*n-1
     def run(i, j, nb_points):
         for ni, nj in [(i-1, j), (i+1, j), (i, j+1), (i, j-1)] :
             if  0<=ni<=m-1 and 0<=nj<=n-1 and grid[ni][nj]==0 and (ni,nj)!=(0,1):
                 grid[ni][nj] = 1
                 run(ni, nj, nb_points+1)
                 grid[ni][nj] = 0
             elif (ni,nj)==(0,1) and nb_points==stop:
                 counter[0] += 1
     run(1, 0, 2)
     return counter[0]
L=[];n=7#maximum for a time < 1 mn
for i in range(2,n):
    for j in range(2,i+1):
       L.append(cycle(i,j))
print(L)

T(m,n) = T(n,m).
T(2m+1,2n+1) = 0.
T(2n,2n) = A003763(n).
T(n,n+1) = T(n+1,n) = A222200(n).
G. functions , G_m(x)=Sum {n>=0} T(m,n-2)*x^n after Stoyan's link p. 18 :
G_2(x) = 1/(1-x) = 1+x+x^2+...
G_3(x) = 1/(1-2*x^2) = 1+2*x^2+4*x^4+...
G_4(x) = 1/(1-2*x-2*x^2+2*x^3-x^4) = 1+2*x+6*x^2+...
G_5(x) = (1+3*x^2)/(1-11*x^2-2*x^6) = 1+14*x^2+154*x^4+...

More terms from Pontus von Brömssen, Feb 15 2021

A003682 Number of (undirected) Hamiltonian paths in the n-ladder graph K_2 X P_n.

Original entry on oeis.org

1, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, 158, 184, 212, 242, 274, 308, 344, 382, 422, 464, 508, 554, 602, 652, 704, 758, 814, 872, 932, 994, 1058, 1124, 1192, 1262, 1334, 1408, 1484, 1562, 1642, 1724, 1808, 1894, 1982, 2072, 2164, 2258, 2354, 2452, 2552, 2654

Offset: 1

Frans J. Faase

Equals row sums of triangle A144336. - Gary W. Adamson, Sep 18 2008

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

Andrew Howroyd, Table of n, a(n) for n = 1..1000
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.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Ladder Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Row n=2 of A332307.
Equals A002061(n) + 1, n > 1.
Cf. A144336. - Gary W. Adamson, Sep 18 2008
Cf. A137882.

a:=n->sum(binomial(2,2*j)+n,j=0..n): seq(a(n), n=0..46); # Zerinvary Lajos, Feb 22 2007
seq(floor((n^3+2*n)/(n+1)),n=1..47); # Gary Detlefs, Feb 20 2010

Join[{1}, Table[n^2 - n + 2, {n, 2, 50}]] (* Harvey P. Dale, Jun 14 2011 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {4, 8, 14}, 50]] (* Harvey P. Dale, Jun 14 2011 *)

a(n)=if(n>1, n^2-n+2, 1) \\ Charles R Greathouse IV, Jan 05 2018

For n>1, a(n) = n^2 - n + 2 = A014206(n-1).
Equals binomial transform of [1, 3, 1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 23 2008
G.f.: x*(1 + x - x^2 + x^3)/(1-x)^3. - R. J. Mathar, Dec 16 2008
a(n) = floor((n^3 + 2*n)/(n+1)). - Gary Detlefs, Feb 20 2010
Except for the first term, a(n) = 2*n + a(n-1), (with a(1)=4). - Vincenzo Librandi, Dec 06 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(1)=1, a(2)=4, a(3)=8. - Harvey P. Dale, Jun 14 2011
Sum_{n>=1} 1/a(n) = 1/2 + Pi*tanh(Pi*sqrt(7)/2)/sqrt(7) = 1.686827... - R. J. Mathar, Apr 24 2024
From Elmo R. Oliveira, Jun 06 2025: (Start)
E.g.f.: exp(x)*(2 + x^2) - (2 + x).
a(n) = A137882(n)/2. (End)

A378938 Array read by antidiagonals: T(m,n) is the number of Hamiltonian paths in an m X n grid which start in the top left corner.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 38, 52, 38, 6, 1, 1, 7, 78, 160, 160, 78, 7, 1, 1, 8, 164, 469, 824, 469, 164, 8, 1, 1, 9, 332, 1337, 3501, 3501, 1337, 332, 9, 1, 1, 10, 680, 3750, 16262, 22144, 16262, 3750, 680, 10, 1

Offset: 1

Andrew Howroyd, Dec 20 2024

These paths are also called Greek-key tours. The path can end anywhere.

			Array begins:
======================================================
m\n | 1 2   3    4     5      6        7         8 ...
----+-------------------------------------------------
  1 | 1 1   1    1     1      1        1         1 ...
  2 | 1 2   3    4     5      6        7         8 ...
  3 | 1 3   8   17    38     78      164       332 ...
  4 | 1 4  17   52   160    469     1337      3750 ...
  5 | 1 5  38  160   824   3501    16262     68591 ...
  6 | 1 6  78  469  3501  22144   144476    899432 ...
  7 | 1 7 164 1337 16262 144476  1510446  13506023 ...
  8 | 1 8 332 3750 68591 899432 13506023 180160012 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..435 (first 29 antidiagonals)

Main diagonal is A145157.
Rows 1..8 are A000012, A000027, A046994, A046995, A145156, A160240, A160241, A374307.
Cf. A332307, A064298, A271465, A271592, A288518.

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

A003695 Number of Hamiltonian paths in P_4 X P_n.

Original entry on oeis.org

1, 14, 62, 276, 1006, 3610, 12010, 38984, 122188, 375122, 1128446, 3342794, 9767588, 28217820, 80709424, 228864620, 644060262, 1800346140, 5002457832, 13825549136, 38026348240, 104133664506, 284037629690, 771953153918, 2091075938320, 5647162827592, 15208169217918

Offset: 1

Frans J. Faase

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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, signature (6,-5,-27,37,48,-69,-38,57,-2,-31,13,3,-4,1).

Row n=4 of A332307.

CoefficientList[Series[1 + 2 x (x^14 - 3 x^13 + 4 x^12 + 10 x^11 - 30 x^10 + 16 x^9 + 36 x^8 - 72 x^7 + 43 x^6 + 67 x^5 - 55 x^4 - 19 x^3 + 13 x^2 + 11 x - 7)/((x^2 + x - 1) (x^4 - 2 x^3 + 2 x^2 + 2 x - 1)^2 (x^4 - x^3 - 3 x^2 - x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 13 2013 *)

a(1) = 1,
a(2) = 14,
a(3) = 62,
a(4) = 276,
a(5) = 1006,
a(6) = 3610,
a(7) = 12010,
a(8) = 38984,
a(9) = 122188,
a(10) = 375122,
a(11) = 1128446,
a(12) = 3342794,
a(13) = 9767588,
a(14) = 28217820,
a(15) = 80709424,
a(16) = 228864620 and
a(n) = 6a(n-1) - 5a(n-2) - 27a(n-3) + 37a(n-4) + 48a(n-5) - 69a(n-6) - 38a(n-7) + 57a(n-8) - 2a(n-9) - 31a(n-10) + 13a(n-11) + 3a(n-12) - 4a(n-13) + a(n-14).
G.f.: x +2*x^2*(x^14 -3*x^13 +4*x^12 +10*x^11 -30*x^10 +16*x^9 +36*x^8 -72*x^7 +43*x^6 +67*x^5 -55*x^4 -19*x^3 +13*x^2 +11*x -7)/((x^2 +x -1) *(x^4 -2*x^3 +2*x^2 +2*x -1)^2 *(x^4 -x^3 -3*x^2 -x +1)). - Colin Barker, Aug 23 2012

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

A003685 Number of Hamiltonian paths in P_3 X P_n.

Original entry on oeis.org

1, 8, 20, 62, 132, 336, 688, 1578, 3190, 6902, 13878, 29038, 58238, 119518, 239390, 485822, 972414, 1960830, 3923326, 7882494, 15768574, 31616510, 63240702, 126655486, 253327358, 507033598, 1014102014, 2029023230, 4058120190, 8118001662, 16236158974, 32476086270

Offset: 1

Frans J. Faase

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..500
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.

Row n=3 of A332307.

a(n) = 3*a(n-1) + 2*a(n-2) - 12*a(n-3) + 4*a(n-4) + 12*a(n-5) - 8*a(n-6), n>8.
From David Bevan, Jul 21 2006: (Start)
a(2*m) = 121*2^(2*m-4) - 4*m*2^m - 25*2^(m-2) - 2, m > 1.
a(2*m+1) = 121*2^(2*m-3) - 31*m*2^(m-2) - 23*2^(m-1) - 2, m > 0.
a(n) = 8*a(n-2) - 20*a(n-4) + 16*a(n-6) + 6, n > 8. (End)
O.g.f.: (2*x^7-8*x^6+12*x^5-2*x^4-2*x^3-6*x^2+5*x+1)*x/((2*x-1)*(-1+2*x^2)^2*(-1+x)). - R. J. Mathar, Dec 05 2007

Terms a(29) and beyond from Andrew Howroyd, Feb 10 2020

A003778 Number of Hamiltonian paths in P_5 X P_n.

Original entry on oeis.org

1, 22, 132, 1006, 4324, 26996, 109722, 602804, 2434670, 12287118, 49852352, 237425498, 969300694, 4434629912, 18203944458, 80978858522, 333840165288, 1456084764388, 6021921661718, 25904211802080

Offset: 1

Frans J. Faase

nonn

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

Sean A. Irvine, Table of n, a(n) for n = 1..200
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.

Row n=5 of A332307.

Faase gives a 36-term linear recurrence on his web page:
a(1) = 1,
a(2) = 22,
a(3) = 132,
a(4) = 1006,
a(5) = 4324,
a(6) = 26996,
a(7) = 109722,
a(8) = 602804,
a(9) = 2434670,
a(10) = 12287118,
a(11) = 49852352,
a(12) = 237425498,
a(13) = 969300694,
a(14) = 4434629912,
a(15) = 18203944458,
a(16) = 80978858522,
a(17) = 333840165288,
a(18) = 1456084764388,
a(19) = 6021921661718,
a(20) = 25904211802080,
a(21) = 107378816068904,
a(22) = 457440612631750,
a(23) = 1899305396852550,
a(24) = 8036345146341508,
a(25) = 33405640842497978,
a(26) = 140677778437397166,
a(27) = 585243342550350368,
a(28) = 2456482541007655088,
a(29) = 10225087180260916062,
a(30) = 42821044456634131964,
a(31) = 178310739623644629736,
a(32) = 745570951093506967610,
a(33) = 3105442902100584328222,
a(34) = 12970906450154764259728,
a(35) = 54035954199253554652658,
a(36) = 225534416271325317632922,
a(37) = 939676160294548239862008,
a(38) = 3920063808158344161168316 and
a(n) = 9a(n-1) + 13a(n-2) - 328a(n-3) + 412a(n-4) + 4606a(n-5)
- 11333a(n-6) - 30993a(n-7) + 116054a(n-8) + 91896a(n-9) - 647749a(n-10)
+ 46716a(n-11) + 2183660a(n-12) - 1288032a(n-13) - 4582138a(n-14) + 4554646a(n-15)
+ 5907135a(n-16) - 8495755a(n-17) - 4382389a(n-18) + 9710124a(n-19) + 1499560a(n-20)
- 7358998a(n-21) + 149939a(n-22) + 4121575a(n-23) - 474900a(n-24) - 1872534a(n-25)
+ 392241a(n-26) + 637672a(n-27) - 187640a(n-28) - 147856a(n-29) + 48980a(n-30)
+ 28332a(n-31) - 13032a(n-32) - 216a(n-33) + 756a(n-34) - 864a(n-35)
+ 432a(n-36).

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

A145402 Number of Hamiltonian paths in P_6 X P_n.

Original entry on oeis.org

1, 32, 336, 3610, 26996, 229348, 1620034, 12071462, 82550864, 572479244, 3808019582, 25304433030, 164452629818, 1062773834046, 6777328517896, 42944798886570, 269706791277978, 1683956271732804, 10445800698724066, 64470330298173718, 395897522698282286

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.

Andrew Howroyd, Table of n, a(n) for n = 1..200
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.

Row n=6 of A332307.

Recurrence:
a(1) = 1,
a(2) = 32,
a(3) = 336,
a(4) = 3610,
a(5) = 26996,
a(6) = 229348,
a(7) = 1620034,
a(8) = 12071462,
a(9) = 82550864,
a(10) = 572479244,
a(11) = 3808019582,
a(12) = 25304433030,
a(13) = 164452629818,
a(14) = 1062773834046,
a(15) = 6777328517896,
a(16) = 42944798886570,
a(17) = 269706791277978,
a(18) = 1683956271732804,
a(19) = 10445800698724066,
a(20) = 64470330298173718,
a(21) = 395897522698282286,
a(22) = 2420749668624155028,
a(23) = 14741571247786709466,
a(24) = 89447754587186752880,
a(25) = 540909580270642216184,
a(26) = 3260975024920004797886,
a(27) = 19603264739475883828250,
a(28) = 117535292246105965344402,
a(29) = 702983297060391275320674,
a(30) = 4195042347314462259387726,
a(31) = 24980876927077036352497846,
a(32) = 148464009996932386776347700,
a(33) = 880707004017612847924259248,
a(34) = 5215420679738577795138490934,
a(35) = 30834760633856575156452382482,
a(36) = 182023498007552212356684065702,
a(37) = 1072972236367114378051620861906,
a(38) = 6316249249418550181323339914312,
a(39) = 37134062572498215721937773361536,
a(40) = 218051132007975699439608964043686,
a(41) = 1278924289541599039994748939762698,
a(42) = 7493036503222763128308036204327090,
a(43) = 43855232912288598091280957567317138,
a(44) = 256423555783154700433887417619421624,
a(45) = 1497918400614505853772957830953728084,
a(46) = 8742417758783236009320473613706164242,
a(47) = 50980753991185396911892104402542597300,
a(48) = 297049767387363496159117043578774571768,
a(49) = 1729483126062016056698341476811920043190,
a(50) = 10061957740464282187277644019379162526042,
a(51) = 58498089362489651097823398471920941376576,
a(52) = 339865477124939798823285486749575905998484,
a(53) = 1973290245189981312766904756242136209547628,
a(54) = 11449989363254903809753791687579863537639720,
a(55) = 66398822904132302559004628977298456048581670,
a(56) = 384828501289828058123250759256477195017480544,
a(57) = 2229130151423292359561588373019497378537925992,
a(58) = 12905482139945922274784040177595268953037073624,
a(59) = 74677955664287358865759062006694983588023954498,
a(60) = 431915003338650359662602332507443189042771688396,
a(61) = 2496891766448143216725256893169977311172853631046,
a(62) = 14427934830066558764818145273279632345264418663372,
a(63) = 83333332226513722399850184075678751393221737658288,
a(64) = 481116428456080286842307490567864574954881424751814,
a(65) = 2776546160822559430889344961278132230852625276213456,
a(66) = 16017287920159426224268234271939994702068236683096952,
a(67) = 92365173104462405690384888989423493983021289807825804,
a(68) = 532437005265425572947418165685557519144407566379788188,
a(69) = 3068133207157035228673454978373479636659816379514577634,
a(70) = 17673852322813372031623824236311245801227744874201505726,
a(71) = 101775693863391958840045017910039901591690632344440430420,
a(72) = 585891711340413211170711537425939102874247508518247861486,
a(73) = 3371750713444109990037815937074468501619571038412857335812,
a(74) = 19398251338784221478821801406177362259804056900563670388806,
a(75) = 111568795166378500936134915873346624423853693744624963980094,
a(76) = 641504617998364195219904173061021504434944205595353347826434,
a(77) = 3687545584633992227002524686539727550037079894386915761864398,
a(78) = 21191373465544351313564008839832091162448835237173224697058876,
a(79) = 121749810823805837552440067819429634654060015970691974416839648,
a(80) = 699307545280466430615312828047674566576438562745475964475819206,
a(81) = 4015706643021649684623778140868657341335861754220230902896008358,
a(82) = 23054334076887448042148612357995502957762056159889516154348493888,
a(83) = 132325303284215702408282792115957397429549544294052046667316933024,
a(84) = 759338970645831460803214242692994927457861759055035612014096168552,
a(85) = 4356458805495707975500370782695432571275910254201456402839379528946,
a(86) = 24988444359124623229107744283670243331720724254595280823991552991342,
a(87) = 143302897934402302882116650096754970142662529653753598056050316770284,
a(88) = 821643145225604646061901571450963815349943846407622019407540341354616,
a(89) = 4710058370878465868959527620867955712709564866281083454929514852175614,
a(90) = 26995186184460869210022072263346128180529395341521512801342492720405190,
a(91) = 154691149154274176889598244154350780798358396944900226522881927956659924,
a(92) = 886269379919108177564957910048500536178199765464663501388525940521397992,
a(93) = 5076789215691537669631156752154537081293123676966123332888421538853542472,
a(94) = 29076191843316870247359219485871781206517693488359111690563685979512648414,
a(95) = 166499432361553419788395309422566612182648297248726066041877141415208791710,
a(96) = 953271470509106369243543177926418983012312059921495414261416813755999417854,
a(97) = 5456959733549075872001836202918114004175794416738296412041775876328443267258,
a(98) = 31233227754487763526217128218054510752349852159351550242516916958065672040014,
a(99) = 178737857335396135203660185992957708646273101994964328871350864581662287530370,
a(100) = 1022707236608978622068432717505248432291457856084068284186568399312410331810432,
a(101) = 5850900383513940954015281710556649941940025405781617483344419093753387423268476,
a(102) = 33468181433150354888869904159114084742899324754034502110186114491065110022122200,
a(103) = 191417198969507319320956593661939446623346523402513085476986313087536811166538340,
a(104) = 1094638153860869625943819331139931221040188338780796056412326567943248472793958802,
a(105) = 6258961737381454735273349796913292077792628144412979236476938336513611161598106484,
a(106) = 35783051128420195492190011308019977156783612836787052747056431871076609691613022114,
a(107) = 204548842309454453799711455219719889854673842730363951318743553233576097299212795442,
a(108) = 1169129062568797296815375785441355037443753860572032657679922002274550424865242854058,
a(109) = 6681512935985943406141450744800377135890211100687009159899691906982317042322945933878,
a(110) = 38179937649795944235517484796055369991364169688382876782534932718852621580273012573744,
a(111) = 218144739304402718284564940871623373450822675202683480252794642639223263633040021474644,
a(112) = 1246247939027939105743088329254213268501907434596141236813634178402005420740542450380628,
a(113) = 7118940481078978742024557769284517384845837781593976384711468911293459232187437799337060,
a(114) = 40661037989804834153982399053378750204939616883988496050793347784222242778432371696180884,
a(115) = 232217375173896510618659626810822796515204095972361739279486086828120095100766924292818294,
a(116) = 1326065718326514761447186285188646030881583149366368223603447347470451333312359990991549570, and
a(n) = 33a(n-1) - 393a(n-2) + 1170a(n-3) + 16754a(n-4) - 164617a(n-5)
+ 168322a(n-6) + 4799822a(n-7) - 23163595a(n-8) - 37721142a(n-9) + 600188299a(n-10)
- 961703543a(n-11) - 7272206245a(n-12) + 30652525711a(n-13) + 27150112504a(n-14) - 406244319529a(n-15)
+ 480827117765a(n-16) + 2953483339807a(n-17) - 8985485328915a(n-18) - 8726841020211a(n-19) + 76359542983674a(n-20)
- 51411687550669a(n-21) - 383142786980539a(n-22) + 769376710831963a(n-23) + 983504604086104a(n-24) - 4703988662134811a(n-25)
+ 1019144283245342a(n-26) + 17567564471258435a(n-27) - 21628609429447372a(n-28) - 39047561134742949a(n-29) + 105510774111014965a(n-30)
+ 21549266915229072a(n-31) - 312479090849851496a(n-32) + 203108186553616885a(n-33) + 603350961560577622a(n-34) - 932935395828098489a(n-35)
- 616494505988563931a(n-36) + 2354671848385377084a(n-37) - 440129521587803560a(n-38) - 4025074369990975795a(n-39) + 3383359137577459958a(n-40)
+ 4524502583073183363a(n-41) - 8084316522568907228a(n-42) - 2000061549048744508a(n-43) + 12710939428078341415a(n-44) - 4333420899536278176a(n-45)
- 14287280072219346302a(n-46) + 12897812849694072664a(n-47) + 10635043132409181759a(n-48) - 20121836247512783757a(n-49) - 2202029990005820642a(n-50)
+ 22530069641124845960a(n-51) - 7891916625415123185a(n-52) - 18920106775493172422a(n-53) + 15668168834118829712a(n-54) + 10967729897465381103a(n-55)
- 18494624437114481188a(n-56) - 2065202418569179366a(n-57) + 16226881294479560421a(n-58) - 4583751833861649976a(n-59) - 10856722405314168245a(n-60)
+ 7442713492418171069a(n-61) + 5123463906533577867a(n-62) - 6977981353490105342a(n-63) - 1007944379242231618a(n-64) + 4832178425594778403a(n-65)
- 966351046903429852a(n-66) - 2583974909058260734a(n-67) + 1371059307640140741a(n-68) + 1025598109986396178a(n-69) - 1054651664720734468a(n-70)
- 224161153417985705a(n-71) + 604947327252110406a(n-72) - 68469700394312381a(n-73) - 269654457078878847a(n-74) + 111988757467772581a(n-75)
+ 87394849743853131a(n-76) - 74501889603770590a(n-77) - 14209663463684077a(n-78) + 34158937071201779a(n-79) - 4582941944236689a(n-80)
- 11444460858858639a(n-81) + 5000095099800696a(n-82) + 2563966731017246a(n-83) - 2451346143506823a(n-84) - 130306682773908a(n-85)
+ 826961146658453a(n-86) - 208781411975348a(n-87) - 184972404092705a(n-88) + 118414958556749a(n-89) + 13754378300437a(n-90)
- 35837701864283a(n-91) + 8178737057414a(n-92) + 5877631567661a(n-93) - 3755468753597a(n-94) - 22088646996a(n-95)
+ 749500012384a(n-96) - 234388451540a(n-97) - 54941696376a(n-98) + 54134588620a(n-99) - 8377519672a(n-100)
- 4771746736a(n-101) + 2428864324a(n-102) - 169609016a(n-103) - 198646044a(n-104) + 72401124a(n-105)
- 3896980a(n-106) - 4402412a(n-107) + 1505256a(n-108) - 152572a(n-109) - 37876a(n-110)
+ 17344a(n-111) - 3248a(n-112) + 336a(n-113) - 16a(n-114).

Terms a(20) and beyond from Andrew Howroyd, Feb 10 2020

cp's OEIS Frontend

A120443 Number of (undirected) Hamiltonian paths in the n X n grid graph.

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A321172 Triangle read by rows: T(m,n) = number of Hamiltonian cycles on m X n grid of points (m >= 2, 2 <= n <= m).

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A003682 Number of (undirected) Hamiltonian paths in the n-ladder graph K_2 X P_n.

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A378938 Array read by antidiagonals: T(m,n) is the number of Hamiltonian paths in an m X n grid which start in the top left corner.

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A003695 Number of Hamiltonian paths in P_4 X P_n.

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A003685 Number of Hamiltonian paths in P_3 X P_n.

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A003778 Number of Hamiltonian paths in P_5 X P_n.

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A145402 Number of Hamiltonian paths in P_6 X P_n.

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