A119789 - OEIS

%I #13 Dec 19 2022 03:26:17

%S 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,

%T 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,

%U 13,14,15,16

%N 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.

%H G. C. Greubel, <a href="/A119789/b119789.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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.

%F From _G. C. Greubel_, Dec 17 2022: (Start)

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

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

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

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

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

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

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

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

%e Triangle begins as:

%e   0;

%e   0, 0;

%e   0, 0, 0;

%e   1, 1, 1, 2;

%e   2, 2, 2, 3, 4;

%e   3, 3, 3, 4, 5, 6;

%e   4, 4, 4, 5, 6, 7, 8;

%e   5, 5, 5, 6, 7, 8, 9, 10;

%t f[n_, k_]= If[n<3, 0, If[k==0, n-2, Floor[Log[GoldenRatio, Fibonacci[n]*Fibonacci[k]]]]];

%t Table[f[n, k], {n,0,12}, {k,0,n}]//Flatten

%t (* Second program *)

%t T[n_, k_]:= T[n, k]= If[n<3, 0, If[k<2, n-2, n+k-4]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 17 2022 *)

%o (Magma)

%o A119789:= func< n,k | n le 2 select 0 else k le 1 select n-2 else n+k-4 >;

%o [A119789(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 17 2022

%o (SageMath)

%o def A119789(n,k):

%o     if (n<3): return 0

%o     elif (k<2): return n-2

%o     else: return n+k-4

%o flatten([[A119789(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Dec 17 2022

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

%K nonn,tabl

%O 0,10

%A _Roger L. Bagula_, Jul 30 2006

%E Edited by _G. C. Greubel_, Dec 17 2022