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

0,10

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A322789 A069904 A164978 * A025424 A216505 A265626

Adjacent sequences: A119786 A119787 A119788 * A119790 A119791 A119792

Keyword

nonn,tabl

Author

Roger L. Bagula, Jul 30 2006

Extensions

Edited by G. C. Greubel, Dec 17 2022

Status

approved