A357724 Triangular array read by rows: T(n,k) = Fib(n) mod Fib(k) for 1 <= k <= n, where Fib(k) = A000045(k).

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

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

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).

Links

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

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

Cf. A000045, A357814.

Sequence in context: A342563 A343633 A174469 * A297934 A112166 A112167

Adjacent sequences: A357721 A357722 A357723 * A357725 A357726 A357727

Keyword

nonn,look,tabl

Author

J. M. Bergot and Robert Israel, Oct 12 2022

Status

approved