A111946 Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Fibonacci(k)), 1 <= k <= n.

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, 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, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 1, 1, 1, 1, 1

Offset

1,6

Comments

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.

Links

T. D. Noe, Rows n = 1..150 of triangle, flattened

P. Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Q. 43 (No. 1, 2005), 3-14.

Formula

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

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

Example

Triangle begins:
  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,  1, 21;
  1, 1, 2, 1, 1, 2,  1,  1, 34;
  1, 1, 1, 1, 5, 1,  1,  1,  1, 55;
  ...

Mathematica

T[ n_, k_] := Fibonacci @ GCD[ n, k] (* Michael Somos, Jul 18 2011 *)

Prog

(PARI) {T(n, k) = fibonacci( gcd( n, k))} /* Michael Somos, Jul 18 2011 */
(Magma) /* As triangle */ [[Gcd(Fibonacci(n), Fibonacci(k)): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Dec 20 2015

Crossrefs

Cf. A000045, A111956, A111957.

Sequence in context: A085476 A124944 A094392 * A175788 A237513 A137844

Adjacent sequences: A111943 A111944 A111945 * A111947 A111948 A111949

Keyword

nonn,tabl

Author

N. J. A. Sloane, Nov 28 2005

Status

approved