A106740 Triangle read by rows: greatest common divisors of pairs of Fibonacci numbers greater than 1: T(n, k) = gcd(Fibonacci(n), Fibonacci(k)).

2, 1, 3, 1, 1, 5, 2, 1, 1, 8, 1, 1, 1, 1, 13, 1, 3, 1, 1, 1, 21, 2, 1, 1, 2, 1, 1, 34, 1, 1, 5, 1, 1, 1, 1, 55, 1, 1, 1, 1, 1, 1, 1, 1, 89, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 233, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 377, 2, 1, 5, 2, 1, 1, 2, 5, 1, 2, 1, 1, 610

Offset: 3

Author

Reinhard Zumkeller, May 15 2005

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  2;
  1, 3;
  1, 1, 5;
  2, 1, 1, 8;
  1, 1, 1, 1, 13;
  1, 3, 1, 1,  1, 21;
  2, 1, 1, 2,  1,  1, 34;
  1, 1, 5, 1,  1,  1,  1, 55;
  1, 1, 1, 1,  1,  1,  1,  1, 89;
  2, 3, 1, 8,  1,  3,  2,  1,  1, 144;
  1, 1, 1, 1,  1,  1,  1,  1,  1,   1, 233;
  1, 1, 1, 1, 13,  1,  1,  1,  1,   1,   1, 377;
  2, 1, 5, 2,  1,  1,  2,  5,  1,   2,   1,   1, 610;
		

Links

• G. C. Greubel, Rows n = 3..52 of the triangle, flattened

Crossrefs

Cf. A000012, A000045, A010125, A060904, A061347, A093148, A131534.

Programs

Mathematica

T[n_, k_]:= GCD[Fibonacci[n], Fibonacci[k]];
Table[T[n, k], {n,3,18}, {k,3,n}]//Flatten (* G. C. Greubel, Sep 11 2021 *)

Sage

def T(n,k): return gcd(fibonacci(n), fibonacci(k))
flatten([[T(n,k) for k in (3..n)] for n in (3..18)]) # G. C. Greubel, Sep 11 2021

Formula

T(n, k) = gcd(A000045(n), A000045(k)) for n >= 3 and 3 <= k <= n.

T(n, 3) = abs(A061347(n)).

T(n, 4) = A093148(n-1).

T(n, n) = A000045(n).

From G. C. Greubel, Sep 11 2021: (Start)

T(n, 3) = A131534(n-2).

T(n, 5) = A060904(n).

T(n, 6) = A010125(n).

T(n, n-1) = T(n, n-2) = A000012(n).

T(n, n-3) = A093148(n-5).

T(n, n-4) = A093148(n-5).

T(n, n-5) = A060904(n-5).

T(n, n-6) = A010125(n-6). (End)