

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 2n2k pieces of 1 X 1 tiles (0<=k<=n).


3



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
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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 nboard (a board with dimensions n X 1) using k (1/2,1/2)fence tiles and 2(nk) halfsquares (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,(nk))th entry of the (1/(1x^2),x/(1x)^2) Riordan array.
(1)^(n+k)*T(n,k) is the (n,(nk))th entry of the (1/(1x^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), 169177.


LINKS

Table of n, a(n) for n=0..67.
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.


FORMULA

G.f.: G=(1t*z)/[(1+t*z)*(1z2*t*z+t^2*z^2)]. G=1/(1g), where g=z+t^2*z^2+2*t*z^2/(1t*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]=(F[n])^2, where F[n] are the Fibonacci polynomials defined by F[0]=F[1]=1, F[n]=F[n1]+tF[n2] for n>=2. They satisfy the recurrence relation P[n]=(1+t)(P[n1]+t*P[n2])t^3*P[n3].
T(n,k) = T(n2,k2)+binomial(2*nk1,2*n2*k1).  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.
Triangle starts:
1;
1;
1,2,1;
1,4,4;
1,6,11,6,1;
1,8,22,24,9;


MAPLE

G:=(1t*z)/(1+t*z)/(1z2*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[2 n  k  1, 2 n  2 k  1]]; Table[T[n, k], {n, 0, 11}, {k, 0, 2 Floor[n/2]}]] // Flatten (* Michael De Vlieger, Jun 24 2020 *)


CROSSREFS

Cf. A007598, A158454.
Sequence in context: A306614 A264336 A322038 * A322115 A294217 A123246
Adjacent sequences: A123518 A123519 A123520 * A123522 A123523 A123524


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Oct 16 2006


STATUS

approved



