A119789 T(n, k) = floor(log_{goldenratio}(Fibonacci(n)*Fibonacci(k))), with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n > 2, triangle read by rows.

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 3, 4, 5, 6, 4, 4, 4, 5, 6, 7, 8, 5, 5, 5, 6, 7, 8, 9, 10, 6, 6, 6, 7, 8, 9, 10, 11, 12, 7, 7, 7, 8, 9, 10, 11, 12, 13, 14, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16

Offset: 0

Author

Roger L. Bagula, Jul 30 2006

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  0;
  0, 0;
  0, 0, 0;
  1, 1, 1, 2;
  2, 2, 2, 3, 4;
  3, 3, 3, 4, 5, 6;
  4, 4, 4, 5, 6, 7, 8;
  5, 5, 5, 6, 7, 8, 9, 10;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000045, A001477, A016789, A033627, A035517, A064455, A140090.

Programs

Magma

A119789:= func< n,k | n le 2 select 0 else k le 1 select n-2 else n+k-4 >;
[A119789(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 17 2022

Mathematica

f[n_, k_]= If[n<3, 0, If[k==0, n-2, Floor[Log[GoldenRatio, Fibonacci[n]*Fibonacci[k]]]]];
Table[f[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<3, 0, If[k<2, n-2, n+k-4]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 17 2022 *)

SageMath

def A119789(n,k):
    if (n<3): return 0
    elif (k<2): return n-2
    else: return n+k-4
flatten([[A119789(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 17 2022

Formula

T(n, k) = floor(log_{goldenratio}(Fibonacci(n)*Fibonacci(k))), with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n > 2.

From G. C. Greubel, Dec 17 2022: (Start)

T(n, k) = n+k-4, with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n >= 3.

T(n, n) = 2*T(n, 0).

T(2*n, n) = 0*[n<2] + A016789(n-2)*[n>1].

T(2*n, n+1) = 3*A001477(n-1), for n > 0.

T(2*n, n-1) = A033627(n) - [n=1].

T(3*n, n) = n*[n<2] + 4*A000027(n-2)*[n>1].

Sum_{k=0..n} T(n, k) = 0*[n<2] + A140090(n-2)*[n>1].

Sum_{k=0..n} (-1)^k * T(n, k) = 0*[n<2] + (-1)^n*A064455(n-2)*[n>1]. (End)

Extensions

Edited by G. C. Greubel, Dec 17 2022