A284130 Hosoya triangle of Jacobsthal type, read by rows.

1, 1, 1, 3, 1, 3, 5, 3, 3, 5, 11, 5, 9, 5, 11, 21, 11, 15, 15, 11, 21, 43, 21, 33, 25, 33, 21, 43, 85, 43, 63, 55, 55, 63, 43, 85, 171, 85, 129, 105, 121, 105, 129, 85, 171, 341, 171, 255, 215, 231, 231, 215, 255, 171, 341, 683, 341, 513, 425, 473, 441, 473, 425, 513, 341, 683

Offset

1,4

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11026, rows 1 <= n <= 150.

Matthew Blair, Rigoberto Flórez, Antara Mukherjee, Matrices in the Hosoya triangle, arXiv:1808.05278 [math.CO], 2018.

R. Florez, R. Higuita and L. Junes, GCD property of the generalized star of David in the generalized Hosoya triangle, J. Integer Seq., 17 (2014), Article 14.3.6, 17 pp.

R. Florez and L. Junes, GCD properties in Hosoya's triangle, Fibonacci Quart. 50 (2012), 163-174.

H. Hosoya, Fibonacci Triangle, The Fibonacci Quarterly, 14;2, 1976, 173-178.

Wikipedia, Hosoya triangle

Formula

T(n,k) = A001045(k)*A001045(n - k + 1), 0 < n, 0 < k <= n.

Example

Triangle begins:
   1;
   1,  1;
   3,  1,  3;
   5,  3,  3,  5;
  11,  5,  9,  5, 11;
  21, 11, 15, 15, 11, 21;
  ...

Mathematica

With[{s = Array[(2^# - (-1)^#)/3 &, 150, 0]}, Table[s[[k]] s[[n - k + 1]], {n, Length@ s}, {k, 2, n - 1}]] // Flatten (* Michael De Vlieger, Nov 14 2018, after Joseph Biberstine at A001045 *)

Prog

(PARI) a(n) = if(n<2, n, a(n - 1) + 2*a(n - 2));
for(n=0, 15, for(k=1, n, print1(a(k) * a(n - k + 1), ", "); ); print(); ); \\ Indranil Ghosh, Mar 29 2017

Crossrefs

Cf. A001045.

Sequence in context: A126637 A110091 A263051 * A005474 A012264 A063198

Adjacent sequences: A284127 A284128 A284129 * A284131 A284132 A284133

Keyword

nonn,tabl

Author

Rigoberto Florez, Mar 20 2017

Status

approved