A239264 -id:A239264 - cp's OEIS frontend

A220644 T(n,k) = number of ways to reciprocally link elements of an n X k array either to themselves or to exactly one king-move neighbor.

1, 2, 2, 3, 10, 3, 5, 40, 40, 5, 8, 172, 369, 172, 8, 13, 728, 3755, 3755, 728, 13, 21, 3096, 37320, 92801, 37320, 3096, 21, 34, 13152, 373177, 2226936, 2226936, 373177, 13152, 34, 55, 55888, 3725843, 53841725, 128171936, 53841725, 3725843, 55888, 55, 89

Offset: 1

R. H. Hardin, Dec 17 2012

Table starts
...1........2............3.................5.....................8
...2.......10...........40...............172...................728
...3.......40..........369..............3755.................37320
...5......172.........3755.............92801...............2226936
...8......728........37320...........2226936.............128171936
..13.....3096.......373177..........53841725............7444342896
..21....13152......3725843........1299348473..........431408410784
..34....55888.....37213728.......31371388772........25014514225856
..55...237472....371654153......757341382671......1450226501771584
..89..1009056...3711809483....18283618480037.....84080327982982848
.144..4287616..37070598992...441397115736816...4874715696405194752
.233.18218688.370232236753.10656083384666537.282621433306639435392

			Some solutions for n=3 k=4 0=self 1=nw 2=n 3=ne 4=w 6=e 7=sw 8=s 9=se (reciprocal directions total 10)
..0..6..4..8....6..4..0..0....8..0..0..0....9..6..4..8....6..4..0..0
..0..7..7..2....8..0..9..7....2..8..8..0....8..1..9..2....0..0..8..8
..3..3..6..4....2..0..3..1....0..2..2..0....2..6..4..1....0..0..2..2

Alois P. Heinz, Table of n, a(n) for n = 1..528 (antidiagonals 1..32) (terms n = 1..180 from R. H. Hardin)

Columns k=1-10 give: A000045(n+1), A052978, A220639, A220640, A220641, A220642, A220643, A243314, A243315, A243316.
Main diagonal is A220638.
Cf. A239264.

A243424 Triangle T(n,k) read by rows of number of ways k domicules can be placed on an n X n square (n >= 0, 0 <= k <= floor(n^2/2)).

Original entry on oeis.org

1, 1, 1, 6, 3, 1, 20, 110, 180, 58, 1, 42, 657, 4890, 18343, 33792, 27380, 7416, 280, 1, 72, 2172, 36028, 362643, 2307376, 9382388, 24121696, 37965171, 34431880, 16172160, 3219364, 170985, 1, 110, 5375, 154434, 2911226, 38049764, 355340561, 2408715568

Offset: 0

Alois P. Heinz, Jun 04 2014

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

The n-th row gives the coefficients of the matching-generating polynomial of the n X n king graph. - Eric W. Weisstein, Jun 20 2017

			T(2,1) = 6:
  +---+  +---+  +---+  +---+  +---+  +---+
  |o-o|  |   |  |o  |  |  o|  |o  |  |  o|
  |   |  |   |  ||  |  |  ||  | \ |  | / |
  |   |  |o-o|  |o  |  |  o|  |  o|  |o  |
  +---+  +---+  +---+  +---+  +---+  +---+
T(2,2) = 3:
  +---+  +---+  +---+
  |o-o|  |o o|  |o o|
  |   |  || ||  | X |
  |o-o|  |o o|  |o o|
  +---+  +---+  +---+
Triangle T(n,k) begins:
  1;
  1;
  1,  6,    3;
  1, 20,  110,   180,     58;
  1, 42,  657,  4890,  18343,   33792,   27380,     7416,      280;
  1, 72, 2172, 36028, 362643, 2307376, 9382388, 24121696, 37965171, ...
  ...

Alois P. Heinz, Rows n = 0..13, flattened
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial

Columns k=0-5 give: A000012, A002943(n-1) for n>0, A243464, A243465, A243466, A243467.
Row sums give A220638.
T(n,floor(n^2/2)) gives A243510.
T(n,floor(n^2/4)) gives A243511.
Cf. A242861 (the same for dominoes), A239264.

b:= proc(n, l) option remember; local d, f, k;
      d:= nops(l)/2; f:=false;
      if n=0 then 1
    elif l[1..d]=[f$d] then b(n-1, [l[d+1..2*d][], true$d])
    else for k to d while not l[k] do od;
         expand(b(n, subsop(k=f, l))+
         `if`(k1 and l[k+d+1],
                            x*b(n, subsop(k=f, k+d+1=f, l)), 0)+
         `if`(k>1 and n>1 and l[k+d-1],
                            x*b(n, subsop(k=f, k+d-1=f, l)), 0)+
         `if`(n>1 and l[k+d], x*b(n, subsop(k=f, k+d=f, l)), 0)+
         `if`(k (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n, [true$(n*2)])):
seq(T(n), n=0..7);

b[n_, l_] := b[n, l] = Module[{d, f, k}, d = Length[l]/2; f = False; Which[ n == 0, 1, l[[1 ;; d]] == Table[f, d], b[n-1, Join[l[[d+1 ;; 2d]], Table[ True, d]]], True, For[k = 1, !l[[k]], k++]; Expand[b[n, ReplacePart[l, k -> f]] + If[k1 && l[[k+d+1]], x*b[n, ReplacePart[l, {k -> f, k + d + 1 -> f}]], 0] + If[k>1 && n>1 && l[[k + d - 1]], x*b[n, ReplacePart[ l, {k -> f, k + d - 1 -> f}]], 0] + If[n>1 && l[[k + d]], x*b[n, ReplacePart[l, {k -> f, k+d -> f}]], 0] + If[k f, k+1 -> f}]], 0]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][
  b[n, Table[True, 2n]]];
Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jun 09 2018, after Alois P. Heinz *)

A239273 Number of domicule tilings of a 2n X 2n square grid.

Original entry on oeis.org

1, 3, 280, 3037561, 3263262629905, 326207195516663381931, 3011882198082438957330143630563, 2565014347691062208319404612723752103028288, 201442620359313683494245316355883565275531844406384955392, 1458834332808489549111708247664894524221330758005874053074138540424018259

Offset: 0

Alois P. Heinz, Mar 13 2014

nonn

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

Number of perfect matchings in the 2n X 2n kings graph. - Andrew Howroyd, Apr 07 2016

			a(1) = 3:
  +---+   +---+   +---+
  |o o|   |o o|   |o-o|
  || ||   | X |   |   |
  |o o|   |o o|   |o-o|
  +---+   +---+   +---+.
a(2) = 280:
  +-------+ +-------+ +-------+ +-------+ +-------+
  |o o o-o| |o o o-o| |o-o o-o| |o o o o| |o o-o o|
  | X     | | X     | |       | | X  | || | \   / |
  |o o o o| |o o o o| |o o o o| |o o o o| |o o o o|
  |   /  || |   / / | ||  X  || |       | ||     ||
  |o o o o| |o o o o| |o o o o| |o-o o o| |o o o o|
  ||    \ | ||     || |       | |     X | | / /   |
  |o o-o o| |o o-o o| |o-o o-o| |o-o o o| |o o o-o|
  +-------+ +-------+ +-------+ +-------+ +-------+ ...

Eric Weisstein's World of Mathematics, King Graph
Wikipedia, King's graph

Even bisection of main diagonal of A239264.

Cf. A004003, A243510, A243424, A220638.

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/2, f = False, k}, Which[n == 0, 1, l[[1 ;; d]] == Array[f &, d], b[n - 1, Join[l[[d + 1 ;; 2*d]], Array[True &, d]]], True, For[k = 1, ! l[[k]], k++]; If[k < d && n > 1 && l[[k + d + 1]], b[n, ReplacePart[l, {k -> f, k + d + 1 -> f}]], 0] + If[k > 1 && n > 1 && l[[k + d - 1]], b[n, ReplacePart[l, {k -> f, k + d - 1 -> f}]], 0] + If[n > 1 && l[[k + d]], b[n, ReplacePart[l, {k -> f, k + d -> f}]], 0] + If[k < d && l[[k + 1]], b[n, ReplacePart[l, {k -> f, k + 1 -> f}]], 0]]];
A[n_, k_] := If[Mod[n*k, 2]>0, 0, If[k>n, A[k, n], b[n, Array[True&, k*2]]]];
a[n_] := A[2n, 2n];
Table[Print[n]; a[n], {n, 0, 7}] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A239264 *)

a(n) = A239264(2n,2n).

a(8) from Alois P. Heinz, Sep 30 2014

a(9) from Alois P. Heinz, Nov 23 2018

A239265 Number of domicule tilings of a 3 X 2n grid.

Original entry on oeis.org

1, 5, 43, 451, 4945, 54685, 605707, 6710971, 74358721, 823915861, 9129240139, 101154812563, 1120826772817, 12419109262381, 137607593744107, 1524734943844939, 16894537473570817, 187196730554444581, 2074198005431257579, 22982759116542299875

Offset: 0

Alois P. Heinz, Mar 13 2014

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

			a(1) = 5:
  +---+  +---+  +---+  +---+  +---+
  |o o|  |o o|  |o-o|  |o-o|  |o-o|
  | X |  || ||  |   |  |   |  |   |
  |o o|  |o o|  |o-o|  |o o|  |o o|
  |   |  |   |  |   |  || ||  | X |
  |o-o|  |o-o|  |o-o|  |o o|  |o o|
  +---+  +---+  +---+  +---+  +---+.

Alois P. Heinz, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (13,-21,-3).

Even bisection of column k=3 of A239264.

gf:= -(x^2+8*x-1)/(3*x^3+21*x^2-13*x+1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

G.f.: -(x^2+8*x-1)/(3*x^3+21*x^2-13*x+1).

A239266 Number of domicule tilings of a 4 X n grid.

Original entry on oeis.org

1, 1, 11, 43, 280, 1563, 9415, 55553, 331133, 1968400, 11716601, 69716257, 414898579, 2469046811, 14693544104, 87442204835, 520375602855, 3096794588441, 18429266069421, 109673987617376, 652678415082545, 3884139865306433, 23114817718082715, 137558073518189643

Offset: 0

Alois P. Heinz, Mar 13 2014

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

			a(2) = 11:
  +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
  |o o| |o o| |o o| |o-o| |o o| |o-o| |o o| |o o| |o-o| |o-o| |o-o|
  | X | | X | | X | |   | || || |   | || || || || |   | |   | |   |
  |o o| |o o| |o o| |o o| |o o| |o-o| |o o| |o o| |o-o| |o-o| |o o|
  |   | |   | |   | | X | |   | |   | |   | |   | |   | |   | || ||
  |o o| |o o| |o-o| |o o| |o o| |o o| |o o| |o-o| |o o| |o-o| |o o|
  | X | || || |   | |   | | X | | X | || || |   | || || |   | |   |
  |o o| |o o| |o-o| |o-o| |o o| |o o| |o o| |o-o| |o o| |o-o| |o-o|
  +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=4 of A239264.

gf:= -(x-1)*(x^3-x^2+5*x-1)/(5*x^6-11*x^5+30*x^4-30*x^3-2*x^2+7*x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

G.f.: -(x-1)*(x^3-x^2+5*x-1)/(5*x^6-11*x^5+30*x^4-30*x^3-2*x^2+7*x-1).

A239267 Number of domicule tilings of a 5 X 2n grid.

Original entry on oeis.org

1, 21, 1563, 162409, 17508475, 1894621633, 205109410835, 22206188455913, 2404176415007051, 260291084969169553, 28180738494571199683, 3051022897700513626745, 330322812747235906893563, 35762812820215620676404385, 3871905699058282397207463923

Offset: 0

Alois P. Heinz, Mar 13 2014

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

Alois P. Heinz, Table of n, a(n) for n = 0..450

Even bisection of column k=5 of A239264.

gf:= -(2048*x^7 -7680*x^6 -25472*x^5 +42048*x^4 -18928*x^3 +2912*x^2 -124*x+1) / (16384*x^8 -58112*x^7 -180608*x^6 +352480*x^5 -201552*x^4 +46976*x^3 -4394*x^2 +145*x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..20);

G.f.: -(2048*x^7 -7680*x^6 -25472*x^5 +42048*x^4 -18928*x^3 +2912*x^2 -124*x+1) / (16384*x^8 -58112*x^7 -180608*x^6 +352480*x^5 -201552*x^4 +46976*x^3 -4394*x^2 +145*x-1).

A239268 Number of domicule tilings of a 6 X n grid.

Original entry on oeis.org

1, 1, 43, 451, 9415, 162409, 3037561, 55263473, 1017093992, 18633949879, 342050825969, 6273663002379, 115107979930355, 2111655465575629, 38740910476086035, 710728644139932355, 13038974254406437397, 239210680096992061776, 4388527184214799104521

Offset: 0

Alois P. Heinz, Mar 13 2014

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

Alois P. Heinz, Table of n, a(n) for n = 0..750

Column k=6 of A239264.

gf:= -(45*x^18 +330*x^17 -3649*x^16 +872*x^15 +13497*x^14 -31638*x^13 +33844*x^12 +87562*x^11 -231307*x^10 -22714*x^9 +206771*x^8 -57002*x^7 -8736*x^6 +7970*x^5 -2193*x^4 -364*x^3 +145*x^2 +10*x-1) /
(585*x^20 +4335*x^19 -47413*x^18 +4273*x^17 +187195*x^16 -352817*x^15 +385178*x^14 +1070602*x^13 -2911442*x^12 -370773*x^11 +2929813*x^10 -729299*x^9 -407618*x^8 +200422*x^7 -19642*x^6 -15983*x^5 +4787*x^4 +563*x^3 -177*x^2 -11*x+1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..20);

G.f.: -(45*x^18 +330*x^17 -3649*x^16 +872*x^15 +13497*x^14 -31638*x^13 +33844*x^12 +87562*x^11 -231307*x^10 -22714*x^9 +206771*x^8 -57002*x^7 -8736*x^6 +7970*x^5 -2193*x^4 -364*x^3 +145*x^2 +10*x-1) / (585*x^20 +4335*x^19 -47413*x^18 +4273*x^17 +187195*x^16 -352817*x^15 +385178*x^14 +1070602*x^13 -2911442*x^12 -370773*x^11 +2929813*x^10 -729299*x^9 -407618*x^8 +200422*x^7 -19642*x^6 -15983*x^5 +4787*x^4 +563*x^3 -177*x^2 -11*x+1).

A239269 Number of domicule tilings of a 7 X 2n grid.

Original entry on oeis.org

1, 85, 55553, 55263473, 57228320561, 59567383578529, 62052716855623473, 64650946142760951261, 67359700036979921768537, 70182277765258094462607893, 73123194329034252403047192825, 76187359457974079841046201710145, 79379928242473326520049884806574585

Offset: 0

Alois P. Heinz, Mar 13 2014

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Even bisection of column k=7 of A239264.

gf:= -(7893703125*x^35 +1178708506875*x^34 -9471431967075*x^33 -25190320844889*x^32 -9539586874311708*x^31 -410493220050893916*x^30 +575920683970775496*x^29 +18726269678802107312*x^28 -29034124354337289144*x^27 -271800359878010634120*x^26 +133177110631012683908*x^25 +3079586993271739345580*x^24 +7730783335738153680196*x^23 -13782583787844763915596*x^22 -24366977853323332846216*x^21 +42038513809989658019568*x^20
-2063678050944576884326*x^19 -12638594920205361440138*x^18 -17386843344014733116586*x^17 +12426575461737923667314*x^16 +1343983627937159538828*x^15 -1998626828626429701652*x^14 +204472622438434512248*x^13 +108140323865267622480*x^12 -35469623048779376672*x^11 +4748719687765155200*x^10 -335752562560949100*x^9 +11627286098346812*x^8 -19234625432244*x^7 -14741830904132*x^6 +600036486728*x^5 -11552831472*x^4 +119161193*x^3 -637033*x^2 +1525*x-1) /
(165767765625*x^36 +24700588841250*x^35 -207544264492950*x^34 -563331132080334*x^33 -200395385497647183*x^32 -8534040529839498708*x^31 +14421739565668843632*x^30 +373620115417467491764*x^29 -641619825956467695364*x^28 -5341798879289372842564*x^27 +3704450681906208094872*x^26 +62112119203321800127524*x^25 +139265952634127843836508*x^24 -281856942688598542445972*x^23 -423329608424574749966944*x^22 +819513105984638655264308*x^21 -131429598068784609902586*x^20 -183950660210880870863984*x^19
-338671775387238895856372*x^18 +266233302665002558298712*x^17 +10903080854445516491318*x^16 -42213214899090813823964*x^15 +6893131124521390078704*x^14 +1965020207232094351100*x^13 -889373505806780285412*x^12 +147961219061817772452*x^11 -13450469929625673736*x^10 +688585418250974364*x^9 -15421722568196676*x^8 -288352000782012*x^7 +30787771291904*x^6 -957729947364*x^5 +15806918761*x^4 -146042386*x^3 +718330*x^2 -1610*x+1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..20);

G.f.: see Maple program.

A239270 Number of domicule tilings of an 8 X n grid.

Original entry on oeis.org

1, 1, 171, 4945, 331133, 17508475, 1017093992, 57228320561, 3263262629905, 185175369431551, 10529540995776143, 598275977865042347, 34004634498887815603, 1932504421503220832048, 109831420296006021851427, 6242000703148139096486777, 354752087455830720672222391

Offset: 0

Alois P. Heinz, Mar 13 2014

nonn

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=8 of A239264.

A239271 Number of domicule tilings of a 9 X 2n grid.

Original entry on oeis.org

1, 341, 1968400, 18633949879, 185175369431551, 1851260737169108297, 18523901518471987018869, 185376808904045560177646408, 1855186116430353424133583769247, 18566115077411836147307357343137943, 185803902034786238482393324889706764945

Offset: 0

Alois P. Heinz, Mar 13 2014

nonn

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Even bisection of column k=9 of A239264.

cp's OEIS Frontend

A220644 T(n,k) = number of ways to reciprocally link elements of an n X k array either to themselves or to exactly one king-move neighbor.

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A243424 Triangle T(n,k) read by rows of number of ways k domicules can be placed on an n X n square (n >= 0, 0 <= k <= floor(n^2/2)).

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A239273 Number of domicule tilings of a 2n X 2n square grid.

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A239265 Number of domicule tilings of a 3 X 2n grid.

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A239266 Number of domicule tilings of a 4 X n grid.

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A239267 Number of domicule tilings of a 5 X 2n grid.

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A239268 Number of domicule tilings of a 6 X n grid.

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A239269 Number of domicule tilings of a 7 X 2n grid.

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