A350110 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + T(n-3,k-1) + T(n-3,k-2) + T(n-3,k-3) - T(n-4,k-3) - T(n-4,k-4) + delta(n,0)*delta(k,0) - delta(n,1)*delta(k,1), T(n

1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 2, 3, 2, 0, 1, 3, 5, 4, 0, 0, 1, 4, 8, 8, 4, 2, 1, 1, 5, 12, 16, 13, 9, 3, 0, 1, 6, 17, 28, 30, 22, 9, 0, 0, 1, 7, 23, 45, 58, 51, 27, 9, 3, 1, 1, 8, 30, 68, 103, 108, 78, 40, 18, 4, 0, 1, 9, 38, 98, 171, 211, 187, 123, 58, 16, 0, 0

Offset: 0

Author

Michael A. Allen, Dec 21 2021

Keywords

• easy
• nonn
• tabl

Comments

This is the m=3 member in the sequence of triangles A007318, A059259, A350110, A350111, A350112 which give the number of tilings of an (n+k) X 1 board using k (1,m-1)-fences and n-k unit square tiles. A (1,g)-fence is composed of two unit square tiles separated by a gap of width g.

It is also the m=3, t=2 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g. - Michael A. Allen, Dec 27 2021

T(3*j+r-k,k) is the coefficient of x^k in (f(j,x))^(3-r)*(f(j+1,x))^r for r=0,1,2 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.

T(n+3-k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 3.

Sum of (n+3)-th antidiagonal (counting initial 1 as the 0th) is A006500(n).

Examples

			Triangle begins:
   1;
   1,   0;
   1,   0,   0;
   1,   1,   1,   1;
   1,   2,   3,   2,   0;
   1,   3,   5,   4,   0,   0;
   1,   4,   8,   8,   4,   2,   1;
   1,   5,  12,  16,  13,   9,   3,   0;
   1,   6,  17,  28,  30,  22,   9,   0,   0;
   1,   7,  23,  45,  58,  51,  27,   9,   3,   1;
   1,   8,  30,  68, 103, 108,  78,  40,  18,   4,   0;
   1,   9,  38,  98, 171, 211, 187, 123,  58,  16,   0,   0;
   1,  10,  47, 136, 269, 382, 399, 310, 176,  64,  16,   4,   1;
		

Links

• Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
• Michael A. Allen, On A Two-Parameter Family of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.9.8.
• Michael A. Allen and Kenneth Edwards, On Two Families of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.7.1.

Crossrefs

Other members of the two-parameter family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350111 (m=4,t=2), A350112 (m=5,t=2), A354665 (m=2,t=3), A354666 (m=2,t=4), A354667 (m=2,t=5), A354668 (m=3,t=3).

Other triangles related to tiling using fences: A123521, A157897, A335964.

Programs

Mathematica

T[n_, k_]:=If[k<0 || n
				

Formula

T(n,0) = 1.

T(n,n) = delta(n mod 3,0).

T(n,1) = n-2 for n>1.

T(3*j-r,3*j-p) = 0 for j>0, p=1,2, and r=1,...,p.

T(3*(j-1)+p,3*(j-1)) = T(3*j,3*j-p) = j^p for j>0 and p=0,1,2,3.

T(3*j+1,3*j-1) = 3*j(j+1)/2 for j>0.

T(3*j+2,3*j-2) = 3*(C(j+2,4) + C(j+1,2)^2) for j>1.

G.f. of row sums: (1-x)/((1-2*x)*(1+x^2-x^3)).

G.f. of antidiagonal sums: (1-x^2)/((1-x-x^2)*(1+x^3-x^6)).

T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=2*k+1 if k>=0.