%I
%S 1,1,0,1,1,0,1,2,0,1,1,3,1,2,0,1,4,3,3,2,0,1,5,6,5,6,0,1,1,6,10,9,12,
%T 3,3,0,1,7,15,16,21,12,6,3,0,1,8,21,27,35,30,14,12,0,1
%N Triangle read by rows, T(n,k) = T(n1,k) + T(n2,k1) + T(n3,k3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n<k,k) = 0.
%C Sum of nth row = A000073(n+2).  _Reinhard Zumkeller_, Jun 25 2009
%C T(n,k) is the number of tilings of an nboard that use k (1/2,1)fences and nk squares. A (1/2,1)fence is a tile composed of two pieces of width 1/2 separated by a gap of width 1. (Result proved in paper by K. Edwards  see the links section.)  _Michael A. Allen_, Apr 28 2019
%C T(n,k) is the (n,nk)th entry in the (1/(1x^3),x*(1+x)/(1x^3)) Riordan array.  _Michael A. Allen_, Mar 11 2021
%H K. Edwards, <a href="http://www.fq.math.ca/Papers1/46_471/Edwards1108.pdf">A Pascallike triangle related to the tribonacci numbers</a>, Fib. Q., 46/47 (2008/2009), 1825.
%H Kenneth Edwards and Michael A. Allen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Allen/edwards2.html">New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile</a>, J. Int. Seq. 24 (2021) Article 21.3.8.
%F T(n,k) = T(n1,k) + T(n2,k1) + T(n3,k3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n<k,k) = 0.
%e First few rows of the triangle are:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 2, 0, 1;
%e 1, 3, 1, 2, 0;
%e 1, 4, 3, 3, 2, 0;
%e 1, 5, 6, 5, 6, 0, 1;
%e 1, 6, 10, 9, 12, 3, 3, 0;
%e 1, 7, 15, 16, 21, 12, 6, 3, 0;
%e 1, 8, 21, 27, 35, 30, 14, 12, 0, 1;
%e ...
%e T(9,3) = 27 = T(8,3) + T(7,2) + T(6,0) = 16 + 10 + 1.
%t T[n_,k_]:=If[n<k  k<0,0,T[n1,k]+T[n2,k1]+T[n3,k3]+KroneckerDelta[n,k,0]]; Flatten[Table[T[n, k],{n,0,9},{k,0,n}]] (* _Michael A. Allen_, Apr 28 2019 *)
%Y Cf. A120415, A006498.
%Y Other triangles related to tiling using fences: A059259, A123521, A335964.
%K nonn,tabl
%O 0,8
%A _Gary W. Adamson_, Mar 08 2009
%E Name clarified by _Michael A. Allen_, Apr 28 2019
%E Definition improved by _Michael A. Allen_, Mar 11 2021
