A284126 - OEIS

%I #32 Dec 07 2019 12:18:28

%S 4,12,12,28,36,28,68,84,84,68,164,204,196,204,164,396,492,476,476,492,

%T 396,956,1188,1148,1156,1148,1188,956,2308,2868,2772,2788,2788,2772,

%U 2868,2308,5572,6924,6692,6732,6724,6732,6692,6924,5572,13452,16716,16156,16252,16236,16236,16252,16156,16716,13452

%N Hosoya triangle of Pell-Lucas type.

%H Michael De Vlieger, <a href="/A284126/b284126.txt">Table of n, a(n) for n = 1..11325</a> rows 1 <= n <= 150.

%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 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), 163--174.

%H H. Hosoya, <a href="http://www.fq.math.ca/Scanned/14-2/hosoya.pdf">Fibonacci Triangle</a>, The Fibonacci Quarterly, 14;2, 1976, 173-178.

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

%F T(n, k)= a(k)*a(n - k + 1), a (n) = 2 a (n - 1) + a (n - 2), a (0) = a (1) = 2; 0 < n, 0 < k <= n.

%e Triangle begins:

%e     4;

%e    12,  12;

%e    28,  36,  28;

%e    68,  84,  84,  68;

%e   164, 204, 196, 204, 164;

%e   ...

%t a[n_]:= a[n]=If[n<2, 2, 2a[n - 1] + a[n - 2]]; Table[a[k] a[n - k + 1], {n, 20}, {k, n}] // Flatten (* _Indranil Ghosh_, Apr 08 2017, edited by _Michael De Vlieger_, Nov 14 2018 *)

%o (PARI) a(n)  = if(n<2, 2, 2*a(n - 1) + a(n - 2));

%o for(n=1, 20, for(k=1, n, print1(a(k)*a(n - k + 1),", ");); print();) \\ _Indranil Ghosh_, Apr 08 2017

%o (Python)

%o def a(n): return 2 if n<2 else 2*a(n - 1) + a(n - 2)

%o for n in range(1, 21): print [a(k)*a(n - k + 1) for k in range(1, n + 1)] # _Indranil Ghosh_, Apr 08 2017

%o (C)

%o #include <stdio.h>

%o int a(int n){

%o     if(n<2){ return 2;}

%o     return 2*a(n - 1) + a(n - 2);

%o }

%o int main()

%o {

%o     int n, k;

%o     for (n=1; n<=20; n++){

%o         for(k=1; k<=n; k++){

%o             printf("%d, ", a(k)*a(n - k + 1));

%o         }

%o     printf("\n");

%o     }

%o     return 0;

%o } // _Indranil Ghosh_, Apr 08 2017

%o (Go)

%o package main

%o import "fmt"

%o func a(n int)int{

%o     if n<2{ return 2 }

%o     return 2*a(n - 1) + a(n - 2)}

%o func main() {

%o    for n:=1; n<=20; n++{

%o        for k:=1; k<=n; k++{

%o            fmt.Printf("%d, ",a(k)*a(n - k + 1))}

%o         fmt.Println()}

%o } // _Indranil Ghosh_, Apr 08 2017

%K nonn,tabl

%O 1,1

%A _Rigoberto Florez_, Mar 20 2017