A108038 Triangle read by rows: g.f. = (x+y+x*y)/((1-x-x^2)*(1-y-y^2)).

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, 23, 25, 25, 23, 29, 13, 21, 47, 37, 41, 39, 41, 37, 47, 21, 34, 76, 60, 66, 64, 64, 66, 60, 76, 34, 55, 123, 97, 107, 103, 105, 103, 107, 97, 123, 55, 89, 199, 157, 173, 167, 169, 169, 167

Offset: 0

Author

N. J. A. Sloane, Jun 01 2005

Keywords

• nonn
• tabl

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).

Examples

			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;
  ...
		

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.

Crossrefs

Cf. A000045, A067331 (row sums).

Programs

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 *)

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)