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 A354668 #15 Dec 27 2022 08:59:39
%S A354668 1,1,0,1,0,0,1,0,0,1,1,0,1,2,0,1,1,3,4,0,0,1,2,5,8,0,0,1,1,3,8,12,0,3,
%T A354668 3,0,1,4,12,18,9,12,9,0,0,1,5,16,27,25,29,27,0,0,1,1,6,21,42,51,66,54,
%U A354668 0,6,4,0,1,7,27,62,95,135,108,36
%N A354668 Triangle read by rows: T(n,k) is the number of tilings of an (n+2*k) X 1 board using k (1,2;3)-combs and n-k squares.
%C A354668 This is the m=3, t=3 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 A354668 T(3*j+r-2*k,k) is the coefficient of x^k in (f(j,x))^(3-r)*(f(j+1,x))^r for r=0,1, where f(n,x) is a Narayana's cows polynomial defined by f(n,x)=f(n-1,x)+x*f(n-3,x)+delta(n,0) where f(n<0,x)=0.
%C A354668 T(n+6-2*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 or 6.
%H A354668 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 A354668 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.
%F A354668 T(n,0) = 1.
%F A354668 T(n,n) = delta(n mod 3,0).
%F A354668 T(n,1) = n-4 for n>3.
%F A354668 T(3*j-r,3*j-p) = 0 for j>0, p=1,2, and r=1-p,...,p.
%F A354668 T(n,2*j) = C(n/2,j)^2 for j>0 and n even and 2*j <= n <= 2*j+8.
%F A354668 T(n,2*j) = C((n-1)/2,j)*C((n+1)/2,j) for j>0 and n odd and 2*j < n < 2*j+8.
%F A354668 T(2*j+3*p,2*j-p) = C(j+3,4)^p for j>0 and p=0,1,2.
%F A354668 G.f. of sums of T(n-2*k,k) over k: (1+x^3-x^4-x^5+x^6-2*x^7-x^8-x^9-2*x^10-x^12-x^13-x^15)/((1-x)*(1+x+x^2)*(1-x-x^3)*(1+3*x^3+7*x^6+9*x^9+7*x^12+3*x^15+x^18)).
%F A354668 T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=4*k+1 if k>=0.
%e A354668 Triangle begins:
%e A354668 1;
%e A354668 1, 0;
%e A354668 1, 0, 0;
%e A354668 1, 0, 0, 1;
%e A354668 1, 0, 1, 2, 0;
%e A354668 1, 1, 3, 4, 0, 0;
%e A354668 1, 2, 5, 8, 0, 0, 1;
%e A354668 1, 3, 8, 12, 0, 3, 3, 0;
%e A354668 1, 4, 12, 18, 9, 12, 9, 0, 0;
%e A354668 1, 5, 16, 27, 25, 29, 27, 0, 0, 1;
%e A354668 1, 6, 21, 42, 51, 66, 54, 0, 6, 4, 0;
%e A354668 1, 7, 27, 62, 95, 135, 108, 36, 30, 16, 0, 0;
%e A354668 ...
%t A354668 f[n_]:=If[n<0, 0, f[n-1]+x*f[n-3]+KroneckerDelta[n,0]]; T[n_, k_]:=Module[{j=Floor[(n+2*k)/3], r=Mod[n+2*k,3]}, Coefficient[f[j]^(3-r)*f[j+1]^r, x, k]]; Flatten@Table[T[n,k], {n, 0, 11}, {k, 0, n}]
%Y A354668 Sums over k of T(n-2*k,k) are A224810.
%Y A354668 Other members of the family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,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).
%Y A354668 Other triangles related to tiling using combs: A059259, A123521, A157897, A335964.
%K A354668 easy,nonn,tabl
%O A354668 0,14
%A A354668 _Michael A. Allen_, Jul 30 2022