A242856 -id:A242856 - cp's OEIS frontend

A242861 Triangle T(n,k) by rows: number of ways k dominoes can be placed on an n X n chessboard, k>=0.

1, 1, 1, 4, 2, 1, 12, 44, 56, 18, 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36, 1, 40, 686, 6632, 39979, 157000, 407620, 695848, 762180, 510752, 192672, 35104, 2180, 1, 60, 1622, 26172, 281514, 2135356, 11785382, 48145820, 146702793, 333518324, 562203148

Offset: 0

Ralf Stephan, May 24 2014

Also, coefficients of the matching-generating polynomial of the n X n grid graph.

In the n-th row there are floor(n^2/2)+1 values.

			Triangle starts:
  1
  1
  1  4   2
  1 12  44   56    18
  1 24 224 1044  2593   3388   2150    552     36
  1 40 686 6632 39979 157000 407620 695848 762180 510752 192672 35104 2180
  ...

Alois P. Heinz, Rows n = 0..14, flattened
Ralf Stephan, Two dominoes on the 3x3 chessboard, illustration of T(3,2)=44.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Wikipedia, Matching polynomial
Index entries for sequences related to dominoes
Index entries for sequences related to matchings

Cf. A046741, A096713.

b:= proc(n, l) option remember; local k;
      if n=0 then 1
    elif min(l[])>0 then b(n-1, map(h->h-1, l))
    else for k while l[k]>0 do od; expand(`if`(n>1,
         x*b(n, subsop(k=2, l)), 0) +`if`(k (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
seq(T(n), n=0..8); # Alois P. Heinz, Jun 01 2014

b[n_, l_List] := b[n, l] =  Module[{k}, Which[n == 0, 1, Min[l]>0, b[n-1, l-1], True, For[k=1, l[[k]]>0, k++]; Expand[If[n>1, x*b[n, ReplacePart[l, k -> 2]], 0] + If[k 1, k + 1 -> 1}]], 0] + b[n, ReplacePart[l, k -> 1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, Array[0&, n]]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 16 2015, after Alois P. Heinz *)

def T(n,k):
   G = Graph(graphs.Grid2dGraph(n,n))
   G.relabel()
   mu = G.matching_polynomial()
   return abs(mu[n^2-2*k])

T(n,1) = A046092(n-1), T(n,2) = A242856(n).
T(n,floor(n^2/2)) = A137308(n), T(2n,2n^2) = A004003(n).
sum(k>=0, T(n,k)) = A210662(n,n) = A028420(n).
T(n,3) = A243206(n), T(n,4) = A243215(n), T(n,5) = A243217(n), T(n,floor(n^2/4)) = A243221(n). - Alois P. Heinz, Jun 01 2014

A344679 Number of 2-matchings of the n-th centered square grid graph.

Original entry on oeis.org

0, 0, 86, 544, 1854, 4688, 9910, 18576, 31934, 51424, 78678, 115520, 163966, 226224, 304694, 401968, 520830, 664256, 835414, 1037664, 1274558, 1549840, 1867446, 2231504, 2646334, 3116448, 3646550, 4241536, 4906494, 5646704, 6467638, 7374960, 8374526, 9472384, 10674774

Offset: 1

Nicolas Bělohoubek, Aug 17 2021

Number of ways two dominoes can be placed on an "other" Aztec Diamonds chessboard.

			For n=1 there is no way to place 2 dominoes in the centered square grid graphs, because they don't have enough space to be placed, so a(1)=0.
For n=2 there is no way to place 2 dominoes in the centered square grid graphs, because the first domino will cover the center square every time, so a(2)=0.

Nicolas Bělohoubek, Visualization of 3rd term.
Ron Knott, 1.2.5 The "other" Aztec Diamonds
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001844, A242856.

a(n) = 2*(n-2)*(4n^3-8n^2+n+4) for n > 1.
From Stefano Spezia, Aug 17 2021: (Start)
G.f.: 2*x^3*(43 + 57*x - 3*x^2 - x^3)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 6. (End)

A348134 Number of ways two L-tiles (with rotation) can be placed on an n X n square.

Original entry on oeis.org

0, 0, 22, 336, 1422, 3952, 8790, 16992, 29806, 48672, 75222, 111280, 158862, 220176, 297622, 393792, 511470, 653632, 823446, 1024272, 1259662, 1533360, 1849302, 2211616, 2624622, 3092832, 3620950, 4213872, 4876686, 5614672, 6433302, 7338240, 8335342, 9430656

Offset: 1

Nicolas Bělohoubek, Oct 02 2021

All terms are even, because groups of ways, which are connected by 90 degrees rotation symmetry, are made up from 4 or 2 ways, so the number of ways will be some 4m+2n, and 4m+2n is even.

			For a(1) and a(2) there are fewer squares on the main square then squares of the 2 L-tiles, so a(1) = a(2) = 0.

Nicolas Bělohoubek, Visualization of 3rd term
Nicolas Bělohoubek, 90° rotation groups for 3rd term
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A242856, A243645.

LinearRecurrence[{5,-10,10,-5,1},{0,0,22,336,1422,3952},40] (* Harvey P. Dale, Mar 04 2023 *)

a(n) = 2*(n - 2)*(4*n^3 - 8*n^2 - 19*n + 32) for n > 1.
G.f.: 2*x^3*(11 + 113*x - 19*x^2 - 9*x^3)/(1 - x)^5. - Stefano Spezia, Oct 03 2021
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Aug 05 2025

A372855 Number of ways two dihexes can be placed on an n-th regular hexagonal board.

Original entry on oeis.org

0, 33, 702, 3630, 11409, 27603, 56748, 104352, 176895, 281829, 427578, 623538, 880077, 1208535, 1621224, 2131428, 2753403, 3502377, 4394550, 5447094, 6678153, 8106843, 9753252, 11638440, 13784439, 16214253, 18951858, 22022202, 25451205, 29265759, 33493728

Offset: 1

Nicolas Bělohoubek, May 15 2024

			Regular hexagonal boards n = 1...4:
. ___
./   \
.\___/
.     ___
. ___/   \___
./   \___/   \
.\___/   \___/
./   \___/   \
.\___/   \___/
.    \___/
.         ___
.     ___/   \___
. ___/   \___/   \___
./   \___/   \___/   \
.\___/   \___/   \___/
./   \___/   \___/   \
.\___/   \___/   \___/
./   \___/   \___/   \
.\___/   \___/   \___/
.    \___/   \___/
.        \___/
.             ___
.         ___/   \___
.     ___/   \___/   \___
. ___/   \___/   \___/   \___
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
.    \___/   \___/   \___/
.        \___/   \___/
.            \___/
For n = 2 the a(2) = 33: (without grid)
. . . . . . . . . . . . . . . . . . .
.   x---x   .   x---x   .   x---x   .
.           .           .           .
. x---x   o . o   x---x . o   o   o .
.           .           .           .
.   o   o   .   o   o   .   x---x   .
. . . . . . . . . . . . . . . . . . .
.   x---x   .   x---x   .   x---x   .
.           .           .           .
. x   o   o . o   x   o . o   x   o .
.  \        .      \    .    /      .
.   x   o   .   o   x   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   x---x   .   o   o   .   o   x   .
.           .           .        \  .
. o   o   x . x---x   o . x---x   x .
.        /  .           .           .
.   o   x   .   x---x   .   o   o   .
. . . . . . . . . . . . . . . . . . .
.   o   o   .   o   o   .   o   o   .
.           .           .           .
. x---x   x . o   x---x . x   x---x .
.        /  .           .  \        .
.   o   x   .   x---x   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   x   o   .   o   x   .
.  /        .    \      .        \  .
. x   x---x . o   x   o . o   o   x .
.           .           .           .
.   o   o   .   x---x   .   x---x   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   o   x   .   x   x   .
.  /        .      /    .    \   \  .
. x   o   o . o   x   o . o   x   x .
.           .           .           .
.   x---x   .   x---x   .   o   o   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   x   o   .   o   x   .
.    \      .    \      .        \  .
. x   x   o . o   x   x . x   o   x .
.  \        .        /  .  \        .
.   x   o   .   o   x   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   o   x   .   x   x   .   o   x   .
.        \  .  /     \  .        \  .
. o   x   x . x   o   x . o   x   x .
.      \    .           .    /      .
.   o   x   .   o   o   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   o   o   .   o   x   .   o   o   .
.           .      /    .           .
. x   x   o . x   x   o . x   o   x .
.  \   \    .  \        .  \     /  .
.   x   x   .   x   o   .   x   x   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   x   x   .   x   o   .
.  /        .  /   /    .  /        .
. x   x   o . x   x   o . x   x   o .
.      \    .           .    /      .
.   o   x   .   o   o   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   o   x   .   o   o   .
.  /        .      /    .           .
. x   o   x . o   x   x . o   x   x .
.        /  .        /  .    /   /  .
.   o   x   .   o   x   .   x   x   .
. . . . . . . . . . . . . . . . . . .

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000384, A242856.

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 33, 702, 3630, 11409, 27603}, 50] (* Paolo Xausa, Aug 28 2024 *)

a(n) = (3/2)*(27*n^4 - 90*n^3 + 78*n^2 + 11*n - 24), for n > 1.
a(n) = 5*a(n - 1) - 10*a(n - 2) + 10*a(n - 3) - 5*a(n - 4) + a(n - 5) for n > 6.
G.f.: 3*x^2*(11 + 179*x + 150*x^2 - 17*x^3 + x^4)/(1 - x)^5.
E.g.f.: 36 - 3*x + 3*exp(x)*(27*x^4 + 72*x^3 - 3*x^2 + 26*x - 24)/2. - Stefano Spezia, Jun 04 2024

A242983 n/2 * (n^3 - 2n^2 - 2n + 5).

Original entry on oeis.org

0, 1, 1, 12, 58, 175, 411, 826, 1492, 2493, 3925, 5896, 8526, 11947, 16303, 21750, 28456, 36601, 46377, 57988, 71650, 87591, 106051, 127282, 151548, 179125, 210301, 245376, 284662, 328483, 377175, 431086, 490576, 556017

Offset: 0

Ralf Stephan, Jun 09 2014

For n>1, number of ways to place two dominoes horizontally on an n X n chessboard.

Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A242856, A242861.

Table[n/2 (n^3-2n^2-2n+5),{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,1,12,58},40] (* Harvey P. Dale, Jul 19 2018 *)

a(n) = A019582(n) + A077414(n-2), n>1.

G.f.: x*(-2*x^3 + 17*x^2 - 4*x + 1) / (1-x)^5.

cp's OEIS Frontend

A242861 Triangle T(n,k) by rows: number of ways k dominoes can be placed on an n X n chessboard, k>=0.

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A344679 Number of 2-matchings of the n-th centered square grid graph.

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A348134 Number of ways two L-tiles (with rotation) can be placed on an n X n square.

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A372855 Number of ways two dihexes can be placed on an n-th regular hexagonal board.

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A242983 n/2 * (n^3 - 2*n^2 - 2*n + 5).

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A242983 n/2 * (n^3 - 2n^2 - 2n + 5).