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%I #78 Nov 22 2022 11:55:23
%S 0,1,1,1,3,1,2,4,4,2,3,7,5,7,3,5,11,9,9,11,5,8,18,14,16,14,18,8,13,29,
%T 23,25,25,23,29,13,21,47,37,41,39,41,37,47,21,34,76,60,66,64,64,66,60,
%U 76,34,55,123,97,107,103,105,103,107,97,123,55,89,199,157,173,167,169,169,167
%N Triangle read by rows: g.f. = (x+y+x*y)/((1-x-x^2)*(1-y-y^2)).
%C Start with 3 rows 0; 1 1; 1 3 1; then rule is each entry is maximum of sum of two entries diagonally above it to the left or to the right. Borders are Fibonacci numbers (A000045).
%H Michael De Vlieger, <a href="/A108038/b108038.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H Matthew Blair, Rigoberto Flórez, Antara Mukherjee, and José L. Ramírez, <a href="https://www.fq.math.ca/Papers1/58-5/blair2.pdf">Matrices in the Determinant Hosoya Triangle</a>, Fibonacci Quart. 58 (2020), no. 5, 34-54.
%H Matthew Blair, Rigoberto Flórez and Antara Mukherjee, <a href="http://math.colgate.edu/~integers/v90/v90.pdf">Geometric Patterns in The Determinant Hosoya Triangle</a>, INTEGERS, A90, 2021.
%H Hsin-Yun Ching, Rigoberto Flórez and Antara Mukherjee, <a href="https://doi.org/10.1515/spma-2020-0116">Families of Integral Cographs within a Triangular Arrays</a>, Special Matrices, 8 (2020), 257-273; see also <a href="https://arxiv.org/abs/2009.02770">arXiv preprint</a>, arXiv:2009.02770 [math.CO], 2020.
%H Hsin-Yun Ching, Rigoberto Flórez, F. Luca, Antara Mukherjee, and J. C. Saunders, <a href="https://arxiv.org/abs/2211.10788">Primes and composites in the determinant Hosoya triangle</a>, arXiv:2211.10788 [math.NT], 2022.
%F From _Rigoberto Florez_, Feb 08 2022: (Start)
%F T(n,k) = F(k+2)*F(n-k+2) - F(k+1)*F(n-k+1), where F(n) = Fibonacci(n) = A000045(n).
%F T(n,k) = F(k)*F(n-k+2) + F(k+1)*F(n-k), where F(n) = Fibonacci(n).
%F T(n,k) = T(n-1,k) + T(n-2,k) and T(n,k) = T(n-1,k-1) + T(n-2,k-2), where T(1,1) = 0, T(2,1) = T(2,2) = 1, and T(3,2) = 3.
%F G.f: (x + x*y + x^2*y)/((1 - x - x^2)*(1 - x*y - x^2*y^2)). (End)
%e Triangle begins:
%e k=0 1 2 3 4
%e n=0: 0;
%e n=1: 1, 1;
%e n=2: 1, 3, 1;
%e n=3: 2, 4, 4, 2;
%e n=4: 3, 7, 5, 7, 3;
%e ...
%t Block[{nn = 11, s}, s = Series[(x + y + x*y)/((1 - x - x^2)*(1 - y - y^2)), {x, 0, nn}, {y, 0, nn}]; Table[Function[m, SeriesCoefficient[s, {m, k}]][n - k], {n, 0, nn}, {k, 0, n}]] // Flatten (* _Michael De Vlieger_, Dec 04 2020 *)
%t G[n_,k_] := Fibonacci[k]*Fibonacci[n-k+1]; T[n_,k_]:= G[n+2,k+1]-G[n,k]; RowPointHosoya[n_] := Table[Inset[T[n,i+1], {1-n+2i,1-n}], {i,0,n-1}]; T[n_] := Graphics[ Flatten[Table[RowPointHosoya[i], {i,1,n}],1]]; Manipulate[T[n], Style["Determinant Hosoya Triangle",12,Red], {{n,6,"Rows"}, Range[12]}, ControlPlacement -> Up] (* _Rigoberto Florez_, Feb 07 2022 *)
%Y Cf. A000045, A067331 (row sums).
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, Jun 01 2005
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