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

0,5

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

Cf. A000045, A067331 (row sums).

Sequence in context: A227147 A074585 A183312 * A151845 A230448 A201653

Adjacent sequences: A108035 A108036 A108037 * A108039 A108040 A108041

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 01 2005

Status

approved