A123521 Triangle read by rows: T(n,k)=number of tilings of a 2 X n grid with k pieces of 1 X 2 tiles (in horizontal position) and 2n-2k pieces of 1 X 1 tiles (0<=k<=n).

1, 1, 1, 2, 1, 1, 4, 4, 1, 6, 11, 6, 1, 1, 8, 22, 24, 9, 1, 10, 37, 62, 46, 12, 1, 1, 12, 56, 128, 148, 80, 16, 1, 14, 79, 230, 367, 314, 130, 20, 1, 1, 16, 106, 376, 771, 920, 610, 200, 25, 1, 18, 137, 574, 1444, 2232, 2083, 1106, 295, 30, 1, 1, 20, 172, 832, 2486, 4744, 5776, 4352, 1897, 420, 36

Offset

0,4

Comments

Also the triangle of the coefficients of the squares of the Fibonacci polynomials. Row n has 1+2*floor(n/2) terms. Sum of terms in row n = (Fibonacci(n+1))^2 (A007598).

From Michael A. Allen, Jun 24 2020: (Start)

T(n,k) is the number of tilings of an n-board (a board with dimensions n X 1) using k (1/2, 1/2)-fence tiles and 2*(n-k) half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal). A (1/2, 1/2)-fence is a tile composed of two 1/2 X 1 pieces separated by a gap of width 1/2.

T(n,k) is the (n, (n-k))-th entry of the (1/(1-x^2), x/(1-x)^2) Riordan array.

(-1)^(n+k)*T(n,k) is the (n, (n-k))-th entry of the (1/(1-x^2), x/(1+x)^2) Riordan array (A158454). (End)

References

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.

Links

G. C. Greubel, Rows n = 0..100 of the irregula triangle, flattened

Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.

Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, arXiv:2009.04649 [math.CO], 2020.

Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, JIS 24 (2021) Article 21.3.8.

Formula

G.f.: G = (1-t*z)/((1+t*z)*(1-z-2*t*z+t^2*z^2)). G = 1/(1-g), where g = z+t^2*z^2+2*t*z^2/(1-t*z) is the g.f. of the indecomposable tilings, i.e., of those that cannot be split vertically into smaller tilings. The row generating polynomials are P(n) = (Fibonacci(n))^2. They satisfy the recurrence relation P(n) = (1+t)*(P(n-1) + t*P(n-2)) - t^3*P(n-3).

T(n,k) = T(n-2,k-2) + binomial(2*n-k-1, 2*n-2*k-1). - Michael A. Allen, Jun 24 2020

Example

T(3,1)=4 because the 1 X 2 tile can be placed in any of the four corners of the 2 X 3 grid.
The irregular triangle begins as:
  1;
  1;
  1,  2,   1;
  1,  4,   4;
  1,  6,  11,   6,    1;
  1,  8,  22,  24,    9;
  1, 10,  37,  62,   46,   12,    1;
  1, 12,  56, 128,  148,   80,   16;
  1, 14,  79, 230,  367,  314,  130,   20,    1;
  1, 16, 106, 376,  771,  920,  610,  200,   25;
  1, 18, 137, 574, 1444, 2232, 2083, 1106,  295,  30,  1;
  1, 20, 172, 832, 2486, 4744, 5776, 4352, 1897, 420, 36;

Maple

G:=(1-t*z)/(1+t*z)/(1-z-2*t*z+t^2*z^2): Gser:=simplify(series(G, z=0, 14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, k), k=0..2*floor(n/2)) od; # yields sequence in triangular form

Mathematica

Block[{T}, T[0, 0]= T[1, 0]= 1; T[n_, k_]:= Which[k==0, 1, k==1, 2(n-1), True, T[n -2, k-2] + Binomial[2n-k-1, 2n-2k-1]]; Table[T[n, k], {n, 0, 14}, {k, 0, 2 Floor[n/2]}]] // Flatten (* Michael De Vlieger, Jun 24 2020 *)

Prog

(Magma)
function A123521(n, k)
  if k eq 0 then return 1;
  elif k eq 1 then return 2*(n-1);
  else return A123521(n-2, k-2) + Binomial(2*n-k-1, 2*n-2*k-1);
  end if; return A123521;
end function;
[A123521(n, k): k in [0..2*Floor(n/2)], n in [0..14]]; // G. C. Greubel, Sep 01 2022
(SageMath)
@CachedFunction
def T(n, k): # T = A123521
    if (k==0): return 1
    elif (k==1): return 2*(n-1)
    else: return T(n-2, k-2) + binomial(2*n-k-1, 2*n-2*k-1)
flatten([[T(n, k) for k in (0..2*(n//2))] for n in (0..12)]) # G. C. Greubel, Sep 01 2022

Crossrefs

Cf. A007598, A158454.

Other triangles related to tiling using fences: A059259, A157897, A335964.

Sequence in context: A264336 A350012 A322038 * A322115 A294217 A123246

Adjacent sequences: A123518 A123519 A123520 * A123522 A123523 A123524

Keyword

nonn,tabf

Author

Emeric Deutsch, Oct 16 2006

Status

approved