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Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Fibonacci(k)), 1 <= k <= n.
3

%I #29 Dec 11 2023 11:44:23

%S 1,1,1,1,1,2,1,1,1,3,1,1,1,1,5,1,1,2,1,1,8,1,1,1,1,1,1,13,1,1,1,3,1,1,

%T 1,21,1,1,2,1,1,2,1,1,34,1,1,1,1,5,1,1,1,1,55,1,1,1,1,1,1,1,1,1,1,89,

%U 1,1,2,3,1,8,1,3,2,1,1,144,1,1,1,1,1,1,1

%N Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Fibonacci(k)), 1 <= k <= n.

%C The function T(n, k) is defined for all integers n, k but only the values for 1 <= k <= n as a triangular array are listed here.

%H T. D. Noe, <a href="/A111946/b111946.txt">Rows n = 1..150 of triangle, flattened</a>

%H P. Ribenboim, <a href="http://www.fq.math.ca/Papers1/43-1/paper43-1-1.pdf">FFF (Favorite Fibonacci Flowers)</a>, Fib. Q. 43 (No. 1, 2005), 3-14.

%F T(n, k) = Fibonacci(gcd(n, k)).

%F T(n, k) = T(k, n) = T(-n, k) = T(n, -k) = T(n, n+k) = T(n+k, k). - _Michael Somos_, Jul 18 2011

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 1, 3;

%e 1, 1, 1, 1, 5;

%e 1, 1, 2, 1, 1, 8;

%e 1, 1, 1, 1, 1, 1, 13;

%e 1, 1, 1, 3, 1, 1, 1, 21;

%e 1, 1, 2, 1, 1, 2, 1, 1, 34;

%e 1, 1, 1, 1, 5, 1, 1, 1, 1, 55;

%e ...

%t T[ n_, k_] := Fibonacci @ GCD[ n, k] (* _Michael Somos_, Jul 18 2011 *)

%o (PARI) {T(n, k) = fibonacci( gcd( n, k))} /* _Michael Somos_, Jul 18 2011 */

%o (Magma) /* As triangle */ [[Gcd(Fibonacci(n), Fibonacci(k)): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Dec 20 2015

%Y Cf. A000045, A111956, A111957.

%K nonn,tabl

%O 1,6

%A _N. J. A. Sloane_, Nov 28 2005