

A357724


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


2



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, 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, 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
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OFFSET

1,14


COMMENTS

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.
If q is even and k is even, or q == 0 (mod 4) and k is odd, T(n,k) = A000045(r).
If q == 1 (mod 4) and r is odd, or q == 3 (mod 4) and r+k is odd, T(n,k) = A000045(kr).
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(kr).


LINKS



EXAMPLE

Triangle starts:
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, 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, 0;


MAPLE

fib:= combinat:fibonacci:
for n from 1 to 20 do
seq(fib(n) mod fib(k), k=1..n)
od;


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



