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%I #34 Sep 02 2022 17:26:28
%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 3,3,0,1,7,15,16,21,12,6,3,0,1,8,21,27,35,30,14,12,0,1,1,9,28,43,57,
%U 61,35,30,6,4,0,1,10,36,65,91,111,81,65,30,10,4,0,1,11,45,94,142,189,169,135,90,30,20,0,1
%N Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n<k,k) = 0.
%C T(n, k) is the number of tilings of an n-board that use k (1/2, 1)-fences and n-k 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, n-k)-th entry in the (1/(1-x^3), x*(1+x)/(1-x^3)) Riordan array. - _Michael A. Allen_, Mar 11 2021
%H G. C. Greubel, <a href="/A157897/b157897.txt">Rows n = 0..50 of the triangle, flattened</a>
%H K. Edwards, <a href="http://www.fq.math.ca/Papers1/46_47-1/Edwards11-08.pdf">A Pascal-like triangle related to the tribonacci numbers</a>, Fib. Q., 46/47 (2008/2009), 18-25.
%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(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n<k,k) = 0.
%F Sum_{k=0..n} T(n, k) = A000073(n+2). - _Reinhard Zumkeller_, Jun 25 2009
%F From _G. C. Greubel_, Sep 01 2022: (Start)
%F T(n, k) = T(n-1, k) + T(n-2, k-1) + T(n-3, k-3), with T(n, 0) = 1.
%F T(n, n) = A079978(n).
%F T(n, n-1) = A087508(n), n >= 1.
%F T(n, 1) = A001477(n-1).
%F T(n, 2) = A161680(n-2).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A120415(n). (End)
%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[n-1,k]+T[n-2,k-1]+T[n-3,k-3]+KroneckerDelta[n,k,0]];
%t Flatten[Table[T[n, k],{n,0,14},{k,0,n}]] (* _Michael A. Allen_, Apr 28 2019 *)
%o (Magma)
%o function T(n,k) // T = A157897
%o if k lt 0 or k gt n then return 0;
%o elif k eq 0 then return 1;
%o else return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3);
%o end if; return T;
%o end function;
%o [T(n,k): k in [0..n], n in [0..14]]; // _G. C. Greubel_, Sep 01 2022
%o (SageMath)
%o def T(n,k): # T = A157897
%o if (k<0 or k>n): return 0
%o elif (k==0): return 1
%o else: return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..14)]) # _G. C. Greubel_, Sep 01 2022
%Y Cf. A000073 (row sums), A006498, A120415.
%Y Cf. A001477, A079978, A087508, A120415, A161680.
%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
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