A284115 - OEIS

%I #41 Jan 20 2025 14:45:59

%S 1,3,3,4,9,4,7,12,12,7,11,21,16,21,11,18,33,28,28,33,18,29,54,44,49,

%T 44,54,29,47,87,72,77,77,72,87,47,76,141,116,126,121,126,116,141,76,

%U 123,228,188,203,198,198,203,188,228,123,199,369,304,329,319,324,319,329,304,369,199

%N Hosoya triangle of Lucas type.

%H Indranil Ghosh, <a href="/A284115/b284115.txt">Rows 1..100, flattened</a>

%H Matthew Blair, Rigoberto Flórez, and 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="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-2/hosoya.pdf">Fibonacci Triangle</a>, The Fibonacci Quarterly, 14;2, 1976, 173-178.

%H Rigoberto 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 Rigoberto 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), 163-174.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hosoya%27s_triangle">Hosoya triangle</a>

%F T(n,k) = L(k)*L(n - k + 1), L(.) is a Lucas number.

%e Triangle begins:

%e    1;

%e    3,  3;

%e    4,  9,  4;

%e    7, 12, 12,  7;

%e   11, 21, 16, 21, 11;

%e   18, 33, 28, 28, 33, 18;

%e   29, 54, 44, 49, 44, 54, 29;

%e   ...

%t Table[LucasL[k] LucasL[n - k + 1] , {n, 10}, {k, n}] // Flatten (* _Indranil Ghosh_, Mar 31 2017 *)

%o (PARI) L(n) = fibonacci(n + 2) - fibonacci(n - 2);

%o for(n=1, 10, for(k=1, n, print1(L(k) * L(n - k + 1),", ");); print();) \\ _Indranil Ghosh_, Mar 31 2017

%o (Python)

%o from sympy import lucas

%o for n in range(1, 11):

%o     print([lucas(k) * lucas(n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Mar 31 2017

%Y Cf. A000032.

%K nonn,tabl

%O 1,2

%A _Rigoberto Florez_, Mar 20 2017