A251072 -id:A251072 - cp's OEIS frontend

A250662 Number A(n,k) of tilings of a 2k X n rectangle using 2n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 6, 36, 1, 1, 1, 1, 1, 1, 13, 95, 1, 1, 1, 1, 1, 1, 7, 22, 281, 1, 1, 1, 1, 1, 1, 1, 15, 64, 781, 1, 1, 1, 1, 1, 1, 1, 8, 25, 155, 2245, 1, 1, 1, 1, 1, 1, 1, 1, 17, 37, 321, 6336, 1, 1

Offset: 0

Alois P. Heinz, Nov 26 2014

			Square array A(n,k) begins:
  1, 1,    1,   1,   1,  1,  1,  1,  1, ...
  1, 1,    1,   1,   1,  1,  1,  1,  1, ...
  1, 1,    5,   1,   1,  1,  1,  1,  1, ...
  1, 1,   11,   6,   1,  1,  1,  1,  1, ...
  1, 1,   36,  13,   7,  1,  1,  1,  1, ...
  1, 1,   95,  22,  15,  8,  1,  1,  1, ...
  1, 1,  281,  64,  25, 17,  9,  1,  1, ...
  1, 1,  781, 155,  37, 28, 19, 10,  1, ...
  1, 1, 2245, 321, 100, 41, 31, 21, 11, ...

Alois P. Heinz, Antidiagonals n = 0..100, flattened
Wikipedia, Polyomino

Columns k=0+1,2-10 give: A000012, A005178(n+1), A236577, A236582, A247117, A250663, A250664, A250665, A250666, A250667.

Cf. A251072.

b:= proc(n, l) option remember; local d, k; d:= nops(l)/2;
      if n=0 then 1
    elif min(l[])>0 then (m->b(n-m, map(x->x-m, l)))(min(l[]))
    else for k while l[k]>0 do od;
         `if`(nd+1 or max(l[k..k+d-1][])>0, 0,
          b(n, [l[1..k-1][],1$d,l[k+d..2*d][]]))
      fi
    end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$2*k])):
seq(seq(A(n,d-n), n=0..d), d=0..14);

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/2, k}, Which[n == 0, 1, Min[l] > 0 , Function[{m}, b[n-m, l-m]][Min[l]], True, For[k=1, l[[k]] > 0, k++]; If[n d]]] + If[d == 1 || k > d+1 || Max[l[[k ;; k+d-1]]] > 0, 0, b[n, Join[l[[1 ;; k-1]], Array[1&, d], l[[k+d ;; 2*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 2k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)

A028468 Number of perfect matchings in graph P_{6} X P_{n}.

Original entry on oeis.org

1, 1, 13, 41, 281, 1183, 6728, 31529, 167089, 817991, 4213133, 21001799, 106912793, 536948224, 2720246633, 13704300553, 69289288909, 349519610713, 1765722581057, 8911652846951, 45005025662792, 227191499132401, 1147185247901449, 5791672851807479

Offset: 0

N. J. A. Sloane, Per H. Lundow

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
R. P. Stanley, Enumerative Combinatorics I, p. 292.

Alois P. Heinz, Table of n, a(n) for n = 0..450
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
David Klarner and Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52.
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
R. J. Mathar, Paving rectangular regions with rectangular tiles: tatami and non-tatami tilings, arXiv:1311.6135 [math.CO], 2013, Table 5.
Thotsaporn "Aek" Thanatipanonda, Statistics of Domino Tilings on a Rectangular Board, Fibonacci Quart. 57 (2019), no. 5, 145-153. See p. 151.
Index entries for linear recurrences with constant coefficients, signature (1,20,10,-38,-10,20,-1,-1).

Row 6 of arrays A099390, A189006.
Column k=2 of A251072.
Cf. A005178.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((1-x^2)*(x^3-5*x^2+6*x-1)*(x^3+ 6*x^2+5*x+1)) )); // G. C. Greubel, Nov 25 2018

seq(coeff(series((1+2*x-x^2)*(x^4+2*x^3-3*x^2-2*x+1)/((x-1)*(x+1)*(x^3-5*x^2+6*x-1)*(x^3+6*x^2+5*x+1)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Nov 23 2018

a[n_] := Product[2(2 + Cos[(2 k Pi)/7] + Cos[(2 j Pi)/(n+1)]), {k, 1, 3}, {j, 1, n/2}] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 19 2018, after A099390 *)
LinearRecurrence[{1, 20, 10, -38, -10, 20, -1, -1}, {1, 1, 13, 41, 281, 1183, 6728, 31529}, 30] (* Vincenzo Librandi, Nov 24 2018 *)

my(x='x+O('x^30)); Vec(-(x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((x-1)*(1+x)*(x^3-5*x^2+6*x-1)*(x^3+6*x^2+5*x+1))) \\ Altug Alkan, Mar 23 2016

s=((x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((1-x^2)*(x^3-5*x^2+6*x-1) *(x^3+6*x^2+5*x+1))).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 25 2018

From N. J. A. Sloane, Feb 03 2009: (Start)
a(1) = 1,
a(2) = 13,
a(3) = 41,
a(4) = 281,
a(5) = 1183,
a(6) = 6728,
a(7) = 31529,
a(8) = 167089,
a(9) = 817991,
a(10) = 4213133,
a(11) = 21001799,
a(12) = 106912793,
a(13) = 536948224,
a(14) = 2720246633, and
a(n) = 40*a(n-2) - 416*a(n-4) + 1224*a(n-6) - 1224*a(n-8) + 416*a(n-10) - 40*a(n-12) + a(n-14). (From Faase's web page.) (End)
G.f.: (x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1) / ( (1-x) *(1+x) *(x^3-5*x^2+6*x-1) *(x^3+6*x^2+5*x+1) ).
a(n) = a(n-1)+20*a(n-2)+10*a(n-3)-38*a(n-4)-10*a(n-5)+20*a(n-6)-a(n-7)-a(n-8). - Sergey Perepechko, Sep 23 2018

A251074 Number of tilings of a 12 X n rectangle using 3n tetrominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 26, 75, 154, 269, 1732, 5764, 15131, 34345, 135950, 462186, 1356284, 3539433, 11681091, 38519022, 118366429, 334591568, 1037603086, 3309045401, 10296063522, 30414763937, 92735958046, 289374852696, 899439481823, 2716896548850, 8270384213984

Offset: 0

Alois P. Heinz, Nov 29 2014

nonn

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

Column k=4 of A251072.

Cf. A236582.

A247218 Number of tilings of a 15 X n rectangle using 3n pentominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 34, 95, 190, 325, 506, 3324, 10353, 25607, 55346, 108756, 389216, 1208901, 3281686, 8006108, 17950204, 51430928, 150609259, 419540401, 1090827453, 2651884943, 7077981621, 19691707908, 54499735145, 145671654672, 371632691473, 976543067070

Offset: 0

Alois P. Heinz, Nov 26 2014

nonn

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

Cf. A174249, A233427, A003520, A247117.

Column k=5 of A251072.

A251073 Number of tilings of a 9 X n rectangle using 3n trominoes of shape I.

Original entry on oeis.org

1, 1, 1, 19, 57, 121, 783, 2861, 8133, 37160, 143419, 468816, 1876855, 7263468, 25496863, 97187247, 372086645, 1352780401, 5071962134, 19220628318, 71025008365, 265095817718, 997839772024, 3713274525679, 13851695644227, 51940567251136, 193830054345968

Offset: 0

Alois P. Heinz, Nov 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alois P. Heinz, G.f. and Maple program for A251073
Wikipedia, Tromino

Column k=3 of A251072.

Cf. A236577.

Maple
```
# see link above.
```

G.f.: see link above.

A251075 Number of tilings of a 18 X n rectangle using 3n hexominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 43, 117, 229, 385, 591, 853, 5789, 17163, 40521, 84211, 160143, 284639, 945352, 2743255, 7017326, 16250983, 34801374, 69959957, 179843441, 479965811, 1238082783, 3015746329, 6925030997, 15065972891, 35869626723, 89673470947, 226122534275

Offset: 0

Alois P. Heinz, Nov 29 2014

nonn

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

Column k=6 of A251072.

Cf. A250663.

A251076 Number of tilings of a 21 X n rectangle using 3n heptominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 53, 141, 271, 449, 681, 973, 1331, 9391, 26803, 60953, 122641, 226921, 394031, 650413, 2039288, 5611209, 13694084, 30417897, 62763334, 122046063, 225958222, 532867801, 1319078776, 3196149101, 7383041514, 16186545853, 33788903208

Offset: 0

Alois P. Heinz, Nov 29 2014

nonn

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

Column k=7 of A251072.

Cf. A250664.

A251077 Number of tilings of a 24 X n rectangle using 3n octominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 64, 167, 316, 517, 776, 1099, 1492, 1961, 14428, 39956, 88103, 172509, 311816, 530678, 860861, 1342433, 4020710, 10599604, 24896626, 53438385, 106915866, 202198270, 365048944, 633709553, 1392777233, 3238633334, 7446642085, 16454751188

Offset: 0

Alois P. Heinz, Nov 29 2014

nonn

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

Column k=8 of A251072.

Cf. A250665.

A251078 Number of tilings of a 27 X n rectangle using 3n nonominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 76, 195, 364, 589, 876, 1231, 1660, 2169, 2764, 21232, 57379, 123291, 235860, 417784, 698657, 1116149, 1717276, 2559760, 7386240, 18797731, 42769684, 89201688, 173879976, 321141337, 567405676, 965768880, 1591873744, 3298800992

Offset: 0

Alois P. Heinz, Nov 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..190
Wikipedia, Nonomino

Column k=9 of A251072.

Cf. A250666.

A251079 Number of tilings of a 30 X n rectangle using 3n decominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 89, 225, 415, 665, 981, 1369, 1835, 2385, 3025, 3761, 30169, 79903, 167957, 314911, 548011, 902339, 1422073, 2161837, 3188141, 4580911, 12815576, 31665861, 70135406, 142741143, 272097694, 492367347, 853768306, 1428360119

Offset: 0

Alois P. Heinz, Nov 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Wikipedia, Decomino

Column k=10 of A251072.

Cf. A250667.

cp's OEIS Frontend

A250662 Number A(n,k) of tilings of a 2k X n rectangle using 2n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A028468 Number of perfect matchings in graph P_{6} X P_{n}.

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A251074 Number of tilings of a 12 X n rectangle using 3n tetrominoes of shape I.

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A247218 Number of tilings of a 15 X n rectangle using 3n pentominoes of shape I.

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A251073 Number of tilings of a 9 X n rectangle using 3n trominoes of shape I.

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Maple

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A251075 Number of tilings of a 18 X n rectangle using 3n hexominoes of shape I.

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A251076 Number of tilings of a 21 X n rectangle using 3n heptominoes of shape I.

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A251077 Number of tilings of a 24 X n rectangle using 3n octominoes of shape I.

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A251078 Number of tilings of a 27 X n rectangle using 3n nonominoes of shape I.

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A251079 Number of tilings of a 30 X n rectangle using 3n decominoes of shape I.

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