A251074 -id:A251074 - cp's OEIS frontend

A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n 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, 13, 1, 1, 1, 1, 1, 41, 1, 1, 1, 1, 1, 19, 281, 1, 1, 1, 1, 1, 1, 57, 1183, 1, 1, 1, 1, 1, 1, 26, 121, 6728, 1, 1, 1, 1, 1, 1, 1, 75, 783, 31529, 1, 1, 1, 1, 1, 1, 1, 34, 154, 2861, 167089, 1, 1, 1, 1, 1, 1, 1, 1, 95, 269, 8133, 817991, 1, 1

Offset: 0

Alois P. Heinz, Nov 29 2014

A(n,n) = A034856(n+2) for n>=2.

			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,     13,    1,    1,   1,   1,   1,  1, ...
  1, 1,     41,   19,    1,   1,   1,   1,  1, ...
  1, 1,    281,   57,   26,   1,   1,   1,  1, ...
  1, 1,   1183,  121,   75,  34,   1,   1,  1, ...
  1, 1,   6728,  783,  154,  95,  43,   1,  1, ...
  1, 1,  31529, 2861,  269, 190, 117,  53,  1, ...
  1, 1, 167089, 8133, 1732, 325, 229, 141, 64, ...

Alois P. Heinz, Antidiagonals n = 0..35, flattened
Wikipedia, Polyomino

Columns k=0+1,2-10 give: A000012, A028468, A251073, A251074, A247218, A251075, A251076, A251077, A251078, A251079.

Cf. A034856, A250662.

b:= proc(n, l) option remember; local d, k; d:= nops(l)/3;
      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`(n2*d+1 or max(l[k..k+d-1][])>0, 0,
          b(n, [l[1..k-1][], 1$d, l[k+d..3*d][]]))
      fi
    end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$3*k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/3, 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 > 2d + 1 || Max[l[[k ;; k + d - 1]]] > 0,  0,  b[n, Join[l[[1 ;; k-1]], Array[1&, d],  l[[k+d ;; 3*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 3k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // 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

A236582 The number of tilings of an 8 X n floor with 1 X 4 tetrominoes.

Original entry on oeis.org

1, 1, 1, 1, 7, 15, 25, 37, 100, 229, 454, 811, 1732, 3777, 7858, 15339, 31273, 65536, 136600, 276535, 562728, 1159942, 2400783, 4918159, 10052140, 20627526, 42480474, 87254743, 178855138, 366854368

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.

R. J. Mathar, Paving rectangular regions..., arXiv:1311.6135 [math.CO], 2013, Table 37.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq. (28).

Cf. A003269 (4 X n floor), A236579 - A236581.
Column k=4 of A250662.
Cf. A251074.

p := (1-x)^3*(x+1)^3*(x^2+1)^3*(x^6-x^4-x^3-x^2+1) ;
q :=  -x^2 -13*x^10 -5*x^18 +8*x^6 -x -x^20 -9*x^4 +16*x^8 -13*x^12 -2*x^19 +1 +10*x^14 +5*x^7 +6*x^15 -6*x^11 +x^22 +6*x^16 +x^17 +2*x^5 -2*x^13 ;
taylor(p/q,x=0,30) ;
gfun[seriestolist](%) ;

G.f.: p(x)/q(x) with polynomials p and q defined in the Maple code.

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A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n 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-2x^2+x^4).

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A236582 The number of tilings of an 8 X n floor with 1 X 4 tetrominoes.

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cp's OEIS Frontend

A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n 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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GAP

Magma

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A236582 The number of tilings of an 8 X n floor with 1 X 4 tetrominoes.

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Maple

Formula

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