OFFSET
0,5
COMMENTS
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).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Matthew Blair, Rigoberto Flórez, Antara Mukherjee, and José L. Ramírez, Matrices in the Determinant Hosoya Triangle, Fibonacci Quart. 58 (2020), no. 5, 34-54.
Matthew Blair, Rigoberto Flórez and Antara Mukherjee, Geometric Patterns in The Determinant Hosoya Triangle, INTEGERS, A90, 2021.
Hsin-Yun Ching, Rigoberto Flórez and Antara Mukherjee, Families of Integral Cographs within a Triangular Arrays, Special Matrices, 8 (2020), 257-273; see also arXiv preprint, arXiv:2009.02770 [math.CO], 2020.
Hsin-Yun Ching, Rigoberto Flórez, F. Luca, Antara Mukherjee, and J. C. Saunders, Primes and composites in the determinant Hosoya triangle, arXiv:2211.10788 [math.NT], 2022.
FORMULA
From Rigoberto Florez, Feb 08 2022: (Start)
T(n,k) = F(k+2)*F(n-k+2) - F(k+1)*F(n-k+1), where F(n) = Fibonacci(n) = A000045(n).
T(n,k) = F(k)*F(n-k+2) + F(k+1)*F(n-k), where F(n) = Fibonacci(n).
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.
G.f: (x + x*y + x^2*y)/((1 - x - x^2)*(1 - x*y - x^2*y^2)). (End)
EXAMPLE
Triangle begins:
k=0 1 2 3 4
n=0: 0;
n=1: 1, 1;
n=2: 1, 3, 1;
n=3: 2, 4, 4, 2;
n=4: 3, 7, 5, 7, 3;
...
MATHEMATICA
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 *)
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jun 01 2005
STATUS
approved