A284129 Hosoya triangle Jacobsthal Lucas type.

1, 5, 5, 7, 25, 7, 17, 35, 35, 17, 31, 85, 49, 85, 31, 65, 155, 119, 119, 155, 65, 127, 325, 217, 289, 217, 325, 127, 257, 635, 455, 527, 527, 455, 635, 257, 511, 1285, 889, 1105, 961, 1105, 889, 1285, 511, 1025, 2555, 1799, 2159, 2015, 2015, 2159, 1799, 2555, 1025, 2047, 5125

Offset

1,2

Links

Indranil Ghosh, Rows 1..100, flattened

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

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

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), pages 163-174.

Wikipedia, Hosoya's triangle.

Formula

T(n,k) = A014551(k)*A014551(n - k + 1), where n > 0 and 0 < k <= n.

Example

Triangle begins:
    1,
    5,    5,
    7,   25,    7,
   17,   35,   35,   17,
   31,   85,   49,   85,   31,
   65,  155,  119,  119,  155,   65,
  127,  325,  217,  289,  217,  325,  127,
  257,  635,  455,  527,  527,  455,  635,  257,
  511, 1285,  889, 1105,  961, 1105,  889, 1285,  511,
  ...

Mathematica

a[n_]:= 2^n + (-1)^n; Table[a[k] a[n - k + 1], {n, 10}, {k, n}] // Flatten (* Indranil Ghosh, Mar 30 2017 *)

Prog

(PARI) a(n) = 2^n + (-1)^n;
for(n=1, 10, for(k=1, n, print1(a(k)*a(n - k + 1), ", "); ); print(); ); \\ Indranil Ghosh, Mar 30 2017
(Python)
def a(n): return 2**n + (-1)**n
for n in range(1, 11):
    print([a(k) * a(n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 30 2017

Crossrefs

Cf. A014551.

Sequence in context: A109257 A088048 A006146 * A077956 A077977 A019204

Adjacent sequences: A284126 A284127 A284128 * A284130 A284131 A284132

Keyword

nonn,tabl

Author

Rigoberto Florez, Mar 20 2017

Status

approved