A357724 - OEIS

%I #32 Oct 23 2022 23:12:54

%S 0,0,0,0,0,0,0,0,1,0,0,0,1,2,0,0,0,0,2,3,0,0,0,1,1,3,5,0,0,0,1,0,1,5,

%T 8,0,0,0,0,1,4,2,8,13,0,0,0,1,1,0,7,3,13,21,0,0,0,1,2,4,1,11,5,21,34,

%U 0,0,0,0,0,4,0,1,18,8,34,55,0,0,0,1,2,3,1,12,2,29,13,55,89,0,0,0,1,2

%N Triangular array read by rows: T(n,k) = Fib(n) mod Fib(k) for 1 <= k <= n, where Fib(k) = A000045(k).

%C For k > 2, T(n,k) = 0 if and only if n is divisible by k. Otherwise, let n = q*k+r with 0 < r < k and k > 2.

%C If q is even and k is even, or q == 0 (mod 4) and k is odd, T(n,k) = A000045(r).

%C If q == 2 (mod 4) and k is odd, T(n,k) = A000045(k) - A000045(r).

%C If q == 1 (mod 4) and r is odd, or q == 3 (mod 4) and r+k is odd, T(n,k) = A000045(k-r).

%C If q == 1 (mod 4) and r is even, or q == 3 (mod 4) and r+k is even, T(n,k) = A000045(k) - A000045(k-r).

%H Robert Israel, <a href="/A357724/b357724.txt">Table of n, a(n) for n = 1..10011</a> (rows 1 to 141, flattened)

%e Triangle starts:

%e   0;

%e   0,  0;

%e   0,  0,  0;

%e   0,  0,  1,  0;

%e   0,  0,  1,  2,  0;

%e   0,  0,  0,  2,  3,  0;

%e   0,  0,  1,  1,  3,  5,  0;

%e   0,  0,  1,  0,  1,  5,  8,  0;

%e   0,  0,  0,  1,  4,  2,  8, 13,  0;

%e   0,  0,  1,  1,  0,  7,  3, 13, 21,  0;

%e   0,  0,  1,  2,  4,  1, 11,  5, 21, 34,  0;

%p fib:= combinat:-fibonacci:

%p for n from 1 to 20 do

%p   seq(fib(n) mod fib(k),k=1..n)

%p od;

%Y Cf. A000045, A357814.

%K nonn,look,tabl

%O 1,14

%A _J. M. Bergot_ and _Robert Israel_, Oct 12 2022