OFFSET
0,12
COMMENTS
T(n,k) is the number of tilings of an n-board (a board with dimensions n X 1) using k (1,1)-fence tiles and n-2k square tiles. A (w,g)-fence tile is composed of two tiles of width w separated by a gap of width g.
Sum of n-th row = A006498(n).
T(2*j+r,k) is the coefficient of x^k in (f(j,x))^(2-r)*(f(j+1,x))^r for r=0,1 where f(n,x) is one form of a Fibonacci polynomial defined by f(n+1,x) = f(n,x) + x*f(n-1,x) where f(0,x)=1 and f(n<0,x)=0. - Michael A. Allen, Oct 02 2021
LINKS
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, J. Int. Seq. 24 (2021) Article 21.3.8.
John Konvalina, On the number of combinations without unit separation., Journal of Combinatorial Theory, Series A 31.2 (1981): 101-107. See Table I.
FORMULA
T(n,k) = A059259(n-k,k).
From Michael A. Allen, Oct 02 2021: (Start)
G.f.: 1/((1 + x^2*y)(1 - x - x^2*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the expansion of the g.f.
T(n,0) = 1.
T(n,1) = n-2 for n>1.
T(n,2) = binomial(n-4,2) + n - 3 for n>3.
T(n,3) = binomial(n-6,3) + 2*binomial(n-5,2) for n>5.
T(4*m-3,2*m-2) = T(4*m-1,2*m-1) = m for m>0.
T(2*n+1,n-k) = A158909(n,k). (End)
T(n,k) = A348445(n-2,k) for n>1.
EXAMPLE
Triangle begins:
1;
1, 0;
1, 0, 0;
1, 1, 0, 0;
1, 2, 1, 0, 0;
1, 3, 2, 0, 0, 0;
1, 4, 4, 0, 0, 0, 0;
1, 5, 7, 2, 0, 0, 0, 0;
1, 6, 11, 6, 1, 0, 0, 0, 0;
1, 7, 16, 13, 3, 0, 0, 0, 0, 0;
1, 8, 22, 24, 9, 0, 0, 0, 0, 0, 0;
1, 9, 29, 40, 22, 3, 0, 0, 0, 0, 0, 0;
...
MATHEMATICA
T[n_, k_]:=If[n<k || k<0, 0, T[n-1, k] + T[n-3, k-1] + T[n-4, k-2] + KroneckerDelta[n, k, 0]]; Flatten[Table[T[n, k], {n, 0, 11}, {k, 0, n}]]
(* or via the g.f.: *)
Flatten[Table[CoefficientList[Series[1/((1+x^2*y)(1 - x - x^2*y)), {x, 0, 23}, {y, 0, 11}], {x, y}][[n+1, k+1]], {n, 0, 11}, {k, 0, n}]]
PROG
(PARI) TT(n, k) = if (n<k, 0, if((n==0) || (k==0), 1, if(k==n, (1+(-1)^n)/2, TT(n-1, k)+TT(n-1, k-1)))) \\ A059259
T(n, k) = TT(n-k, k);
\\ matrix(7, 7, n, k, T(n-1, k-1)) \\ Michel Marcus, Jul 18 2020
CROSSREFS
KEYWORD
AUTHOR
Michael A. Allen, Jul 01 2020
STATUS
approved