This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350110 #31 Dec 27 2022 11:06:19
%S A350110 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,
%T A350110 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,
%U A350110 108,78,40,18,4,0,1,9,38,98,171,211,187,123,58,16,0,0
%N 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<k,k) = T(n,k<0) = 0.
%C A350110 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.
%C A350110 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
%C A350110 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.
%C A350110 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.
%C A350110 Sum of (n+3)-th antidiagonal (counting initial 1 as the 0th) is A006500(n).
%H A350110 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 A350110 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 A350110 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 A350110 T(n,0) = 1.
%F A350110 T(n,n) = delta(n mod 3,0).
%F A350110 T(n,1) = n-2 for n>1.
%F A350110 T(3*j-r,3*j-p) = 0 for j>0, p=1,2, and r=1,...,p.
%F A350110 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.
%F A350110 T(3*j+1,3*j-1) = 3*j(j+1)/2 for j>0.
%F A350110 T(3*j+2,3*j-2) = 3*(C(j+2,4) + C(j+1,2)^2) for j>1.
%F A350110 G.f. of row sums: (1-x)/((1-2*x)*(1+x^2-x^3)).
%F A350110 G.f. of antidiagonal sums: (1-x^2)/((1-x-x^2)*(1+x^3-x^6)).
%F A350110 T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=2*k+1 if k>=0.
%e A350110 Triangle begins:
%e A350110 1;
%e A350110 1, 0;
%e A350110 1, 0, 0;
%e A350110 1, 1, 1, 1;
%e A350110 1, 2, 3, 2, 0;
%e A350110 1, 3, 5, 4, 0, 0;
%e A350110 1, 4, 8, 8, 4, 2, 1;
%e A350110 1, 5, 12, 16, 13, 9, 3, 0;
%e A350110 1, 6, 17, 28, 30, 22, 9, 0, 0;
%e A350110 1, 7, 23, 45, 58, 51, 27, 9, 3, 1;
%e A350110 1, 8, 30, 68, 103, 108, 78, 40, 18, 4, 0;
%e A350110 1, 9, 38, 98, 171, 211, 187, 123, 58, 16, 0, 0;
%e A350110 1, 10, 47, 136, 269, 382, 399, 310, 176, 64, 16, 4, 1;
%t A350110 T[n_, k_]:=If[k<0 || n<k, 0, 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] + KroneckerDelta[n, k, 0] - KroneckerDelta[n, k, 1]]; Flatten[Table[T[n,k], {n, 0, 11}, {k, 0, n}]]
%Y A350110 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).
%Y A350110 Other triangles related to tiling using fences: A123521, A157897, A335964.
%K A350110 easy,nonn,tabl
%O A350110 0,12
%A A350110 _Michael A. Allen_, Dec 21 2021