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%I #47 Sep 19 2024 19:39:26
%S 1,5,5,7,25,7,17,35,35,17,31,85,49,85,31,65,155,119,119,155,65,127,
%T 325,217,289,217,325,127,257,635,455,527,527,455,635,257,511,1285,889,
%U 1105,961,1105,889,1285,511,1025,2555,1799,2159,2015,2015,2159,1799,2555,1025,2047,5125
%N Hosoya triangle Jacobsthal Lucas type.
%H Indranil Ghosh, <a href="/A284129/b284129.txt">Rows 1..100, flattened</a>
%H Matthew Blair, Rigoberto Flórez, Antara Mukherjee, <a href="https://arxiv.org/abs/1808.05278">Matrices in the Hosoya triangle</a>, arXiv:1808.05278 [math.CO], 2018.
%H H. Hosoya, <a href="http://www.fq.math.ca/Scanned/14-2/hosoya.pdf">Fibonacci Triangle</a>, The Fibonacci Quarterly, 14;2, 1976, pages 173-178.
%H R. Florez, R. Higuita and L. Junes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Florez/florez3.html">GCD property of the generalized star of David in the generalized Hosoya triangle</a>, J. Integer Seq., 17 (2014), Article 14.3.6, 17 pp.
%H R. Florez and L. Junes, <a href="http://leandrojunes.com/wp-content/uploads/2014/07/FlorezJunes.pdf">GCD properties in Hosoya's triangle</a>, Fibonacci Quart. 50 (2012), pages 163-174.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hosoya%27s_triangle">Hosoya's triangle</a>.
%F T(n,k) = A014551(k)*A014551(n - k + 1), where n > 0 and 0 < k <= n.
%e Triangle begins:
%e 1,
%e 5, 5,
%e 7, 25, 7,
%e 17, 35, 35, 17,
%e 31, 85, 49, 85, 31,
%e 65, 155, 119, 119, 155, 65,
%e 127, 325, 217, 289, 217, 325, 127,
%e 257, 635, 455, 527, 527, 455, 635, 257,
%e 511, 1285, 889, 1105, 961, 1105, 889, 1285, 511,
%e ...
%t a[n_]:= 2^n + (-1)^n; Table[a[k] a[n - k + 1], {n, 10}, {k, n}] // Flatten (* _Indranil Ghosh_, Mar 30 2017 *)
%o (PARI) a(n) = 2^n + (-1)^n;
%o for(n=1, 10, for(k=1, n, print1(a(k)*a(n - k + 1),", ");); print();); \\ _Indranil Ghosh_, Mar 30 2017
%o (Python)
%o def a(n): return 2**n + (-1)**n
%o for n in range(1, 11):
%o print([a(k) * a(n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Mar 30 2017
%Y Cf. A014551.
%K nonn,tabl
%O 1,2
%A _Rigoberto Florez_, Mar 20 2017