A350111 - OEIS

%I #19 Jan 20 2023 16:47:43

%S 1,1,0,1,0,0,1,0,0,0,1,1,1,1,1,1,2,3,4,2,0,1,3,6,7,4,0,0,1,4,9,12,8,0,

%T 0,0,1,5,13,20,16,8,4,2,1,1,6,18,32,36,28,19,12,3,0,1,7,24,50,69,69,

%U 58,31,9,0,0,1,8,31,74,120,144,127,78,27,0,0,0

%N Triangle read by rows: T(n,k) is the number of tilings of an (n+k)-board using k (1,3)-fences and n-k squares.

%C This is the m=4 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.

%C It is also the m=4, 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.

%C T(4*j+r-k,k) is the coefficient of x^k in (f(j,x))^(4-r)*(f(j+1,x))^r for r=0,1,2,3 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.

%C T(n+4-k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 4.

%C Sum of (n+3)-th antidiagonal (counting initial 1 as the 0th) is A031923(n).

%H Michael A. Allen, <a href="https://arxiv.org/abs/2209.01377">On a Two-Parameter Family of Generalizations of Pascal's Triangle</a>, arXiv:2209.01377 [math.CO], 2022.

%H Michael A. Allen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Allen2/allen8.html">On A Two-Parameter Family of Generalizations of Pascal's Triangle</a>, J. Int. Seq. 25 (2022) Article 22.9.8.

%H Michael A. Allen and Kenneth Edwards, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Allen/allen3.html">On Two Families of Generalizations of Pascal's Triangle</a>, J. Int. Seq. 25 (2022) Article 22.7.1.

%F T(n,k) = T(n-1,k) + T(n-2,k-1) - T(n-3,k-1) + T(n-3,k-2) + T(n-4,k-1) + T(n-4,k-3) + 2*T(n-4,k-4) + T(n-5,k-2) + 2*T(n-5,k-3) - T(n-5,k-4) - T(n-6,k-3)-T(n-6,k-5) - T(n-7,k-4)-T(n-7,k-5) - T(n-7,k-6) - T(n-8,k-7)-T(n-8,k-8) + delta(n,0)*delta(k,0) - delta(n,2)*delta(k,1) - delta(n,3)*delta(k,2) - delta(n,4)*delta(k,4) with T(n<k,k)=T(n,k<0)=0.

%F T(n,0) = 1.

%F T(n,n) = delta(n mod 4,0).

%F T(n,1) = n-3 for n>2.

%F T(4*j-r,4*j-p) = 0 for j>0, p=1,2,3, and r=1,...,p.

%F T(4*(j-1)+p,4*(j-1)) = T(4*j,4*j-p) = j^p for j>0 and p=0,1,2,3,4.

%F T(4*j+1,4*j-1) = 4*j(j+1)/2 for j>0.

%F T(4*j+2,4*j-2) = 4*C(j+2,4) + 6*C(j+1,2)^2 for j>1.

%F G.f. of row sums: (1-x-x^3)/((1-2*x)*(1-x^2)*(1+2*x^2+x^3+x^4)).

%F G.f. of antidiagonal sums: (1-x^2-x^3+x^4-x^6)/((1-x-x^2)*(1-x^4)*(1+3*x^4+x^8)).

%F T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=3*k+1 if k>=0.

%e Triangle begins:

%e   1;

%e   1,   0;

%e   1,   0,   0;

%e   1,   0,   0,   0;

%e   1,   1,   1,   1,   1;

%e   1,   2,   3,   4,   2,   0;

%e   1,   3,   6,   7,   4,   0,   0;

%e   1,   4,   9,  12,   8,   0,   0,   0;

%e   1,   5,  13,  20,  16,   8,   4,   2,   1;

%e   1,   6,  18,  32,  36,  28,  19,  12,   3,   0;

%e   1,   7,  24,  50,  69,  69,  58,  31,   9,   0,   0;

%e   1,   8,  31,  74, 120, 144, 127,  78,  27,   0,   0,   0;

%e   1,   9,  39, 105, 195, 264, 265, 189,  81,  27,   9,   3,   1;

%e   1,  10,  48, 144, 300, 458, 522, 432, 270, 132,  58,  24,   4,   0;

%t f[n_]:=If[n<0,0,f[n-1]+x*f[n-2]+KroneckerDelta[n,0]];

%t T[n_, k_]:=Module[{j=Floor[(n+k)/4],r=Mod[n+k,4]},

%t   Coefficient[f[j]^(4-r)*f[j+1]^r,x,k]];

%t Flatten@Table[T[n,k], {n, 0, 13}, {k, 0, n}]

%t (* or *)

%t T[n_,k_]:=If[k<0 || n<k, 0, T[n-1, k] + T[n-2, k-1] - T[n-3, k-1] + T[n-3, k-2] + T[n-4, k-1] + T[n-4, k-3] + 2*T[n-4, k-4] + T[n-5, k-2] + 2*T[n-5, k-3] - T[n-5, k-4] - T[n-6, k-3] - T[n-6, k-5] - T[n-7, k-4] - T[n-7, k-5] - T[n-7, k-6] - T[n-8, k-7] - T[n-8, k-8] + KroneckerDelta[n, k, 0] - KroneckerDelta[n,2]*KroneckerDelta[k,1] - KroneckerDelta[n,3]*KroneckerDelta[k,2] - KroneckerDelta[n, k, 4]]; Flatten@Table[T[n, k], {n, 0, 9}, {k, 0, n}]

%Y Sums of antidiagonals: A031923

%Y Other members of the two-parameter family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,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).

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

%K easy,nonn,tabl

%O 0,17

%A _Michael A. Allen_, Dec 22 2021