A236582 -id:A236582 - 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 *)

A052535 Expansion of (1-x)(1+x)/(1-x-2x^2+x^4).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 50, 95, 181, 345, 657, 1252, 2385, 4544, 8657, 16493, 31422, 59864, 114051, 217286, 413966, 788674, 1502555, 2862617, 5453761, 10390321, 19795288, 37713313, 71850128, 136886433, 260791401, 496850954, 946583628

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) = number of compositions of n with parts in {2,1,3,5,7,9,...}. The generating function follows easily from Theorem 1.1 of the Hoggatt et al. reference. Example: a(4)= 7 because we have 22, 31, 13, 211, 121, 112, and 1111. - Emeric Deutsch, Aug 17 2016.
Diagonal sums of A054142. - Paul Barry, Jan 21 2005
Equals INVERT transform of (1, 1, 1, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 27 2009
Number of tilings of a 4 X 2n rectangle by 4 X 1 tetrominoes. - M. Poyraz Torcuk, Dec 10 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024. See p. 18.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 465
Todd Mullen, On Variants of Diffusion, Dalhousie University (Halifax, NS Canada, 2020).
Todd Mullen, Richard Nowakowski, and Danielle Cox, Counting Path Configurations in Parallel Diffusion, arXiv:2010.04750 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

Cf. A275446.
Bisection of A003269 (odd part),
Cf. A236582, A251074, A236580.

a:=[1,1,2,4];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-4]; od; a; # G. C. Greubel, May 09 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)/( 1-x-2*x^2+x^4) )); // G. C. Greubel, May 09 2019

spec := [S,{S=Sequence(Prod(Z,Union(Z,Sequence(Prod(Z,Z)))))},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

CoefficientList[Series[(1-x^2)/(1-x-2x^2+x^4), {x, 0, 40}], x] (* or *)
Table[Length@ Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {_, a_, _} /; And[EvenQ@ a, a != 2]]], 1], {n, 0, 40}]  (* Michael De Vlieger, Aug 17 2016 *)
LinearRecurrence[{1,2,0,-1},{1,1,2,4},40] (* Harvey P. Dale, Apr 12 2018 *)

my(x='x+O('x^40)); Vec((1-x^2)/(1-x-2*x^2+x^4)) \\ G. C. Greubel, May 09 2019

((1-x^2)/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

G.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4).
a(n) = a(n-1) + 2*a(n-2) - a(n-4), with a(0)=1, a(1)=1, a(2)=2, a(3)=4.
a(n) = Sum_{alpha = RootOf(1-x-2*x^2+x^4)} (1/283)*(27 + 112*alpha + 9*alpha^2 -48*alpha^3)*alpha^(-n-1).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k, k). - Paul Barry, Jan 21 2005
a(n) = A158943(n) -A158943(n-2). - R. J. Mathar, Jan 13 2023

More terms from James Sellers, Jun 05 2000

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.

A236579 The number of tilings of a 5 X (4n) floor with 1 X 4 tetrominoes.

Original entry on oeis.org

1, 3, 15, 75, 371, 1833, 9057, 44753, 221137, 1092699, 5399327, 26679563, 131831075, 651413681, 3218814561, 15905050017, 78591236385, 388340962771, 1918899743823, 9481812581835, 46852249642771

Offset: 0

R. J. Mathar, Jan 29 2014

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

Related to A002378 by an Invert Transform.

Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 34.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], eq. (24).
Index entries for linear recurrences with constant coefficients, signature (6,-6,4,-1).

Cf. A003269 (4Xn floor), A236580 - A236582, A109960.

g := (1-x)^3/(-6*x+1+6*x^2-4*x^3+x^4) ;
taylor(%,x=0,30) ; gfun[seriestolist](%) ;
# Alternatively:
a := n -> hypergeom([(n+1)/3, (n+2)/3, n/3 + 1, -n], [1/4, 1/2, 3/4], -27/128):
seq(simplify(a(n)), n=0..20); # Peter Luschny, Nov 02 2017

LinearRecurrence[{6, -6, 4, -1}, {1, 3, 15, 75}, 21] (* Jean-François Alcover, Jul 14 2018 *)

G.f.: (1-x)^3/(-6*x+1+6*x^2-4*x^3+x^4).
a(n) = Sum_{k = 0..n} binomial(n + 3*k, 4*k)*2^k = Sum_{k = 0..n} A109960(n,k)*2^k. - Peter Bala, Nov 02 2017
a(n) = hypergeom([(n+1)/3, (n+2)/3, n/3 + 1, -n], [1/4, 1/2, 3/4], -27/128). - Peter Luschny, Nov 02 2017

A236580 The number of tilings of a 6 X (4n) floor with 1 X 4 tetrominoes.

Original entry on oeis.org

1, 4, 25, 154, 943, 5773, 35344, 216388, 1324801, 8110882, 49657576, 304020556, 1861317163, 11395616227, 69767835259, 427142397547, 2615110919020, 16010597772667, 98022320649478, 600125959188877, 3674175070596919, 22494548423870416, 137719270059617428

Offset: 0

R. J. Mathar, Jan 29 2014

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.
R. J. Mathar, Paving rectangular regions..., arXiv:1311.6135, Table 35.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], eq. (26).
Index entries for linear recurrences with constant coefficients, signature (7,-6,4,-1).

Cf. A003269 (4Xn floor), A236579 - A236582.

g := (1-x)^3/(-7*x+1+6*x^2-4*x^3+x^4) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

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

A236581 The number of tilings of a 7 X (4n) floor with 1 X 4 tetrominoes.

Original entry on oeis.org

1, 5, 37, 269, 1949, 14121, 102313, 741305, 5371097, 38916077, 281964941, 2042966149, 14802232757, 107249008849, 777068573905, 5630220503025, 40793546383409, 295568073335893, 2141527121824885, 15516352499614333, 112423136012925517, 814557513519681785

Offset: 0

R. J. Mathar, Jan 29 2014

nonn

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 36.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], eq. (27).
Index entries for linear recurrences with constant coefficients, signature (8,-6,4,-1).

Cf. A003269 (4Xn floor), A236579 - A236582.

g := (1-x)^3/(-8*x+1+6*x^2-4*x^3+x^4) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

LinearRecurrence[{8, -6, 4, -1}, {1, 5, 37, 269}, 19] (* Jean-François Alcover, Feb 19 2019 *)

G.f.: (1-x)^3/(-8*x+1+6*x^2-4*x^3+x^4).

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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A052535 Expansion of (1-x)*(1+x)/(1-x-2*x^2+x^4).

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

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A236579 The number of tilings of a 5 X (4n) floor with 1 X 4 tetrominoes.

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A236580 The number of tilings of a 6 X (4n) floor with 1 X 4 tetrominoes.

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A236581 The number of tilings of a 7 X (4n) floor with 1 X 4 tetrominoes.

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A052535 Expansion of (1-x)(1+x)/(1-x-2x^2+x^4).