A357814 Triangular array read by rows: T(n,k) is the quotient on division of Fib(n) by Fib(k) for 1 <= k <= n, where Fib(k) = A000045(k).

1, 1, 1, 2, 2, 1, 3, 3, 1, 1, 5, 5, 2, 1, 1, 8, 8, 4, 2, 1, 1, 13, 13, 6, 4, 2, 1, 1, 21, 21, 10, 7, 4, 2, 1, 1, 34, 34, 17, 11, 6, 4, 2, 1, 1, 55, 55, 27, 18, 11, 6, 4, 2, 1, 1, 89, 89, 44, 29, 17, 11, 6, 4, 2, 1, 1, 144, 144, 72, 48, 28, 18, 11, 6, 4, 2, 1, 1, 233, 233, 116, 77, 46, 29, 17, 11

Offset

1,4

Links

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

Formula

T(n,k) = floor(A000045(n)/A000045(k)).

A000045(n) = T(n,k)*A000045(k) + A357724(n,k).

The following follow from the identity A000045(n) = A000045(k)*A000032(n-k) + (-1)^k*A000045(n-2*k).

T(2*m*k,k) = Sum_{j=0..m-1} (-1)^((j+m+1)*k)*A000032((2*j+1)*k).

T((2*m+1)*k,k) = -1 + Sum_{j=0..m} (-1)^((j+m)*k)*A000032(2*j*k) if k is even.

T((2*m+1)*k,k) = (-1)^(m+1) + Sum_{j=0..m} (-1)^((j+m)*k)*A000032(2*j*k) if k is odd.

If 2*m*k < n < (2*m+1)*k and k >= 4 is even, then T(n,k) = Sum_{j=0..m-1} A000032(n-(2*j+1)*k)

If 2*m*k < n < (2*m+1)*k, m is even and k is odd, then T(n,k) = Sum_{j=0..m-1} (-1)^j*A000032(n-(2*j+1)*k).

If 2*m*k < n < (2*m+1)*k, m is odd and k is odd, then T(n,k) = -1 + Sum_{j=0..m-1} (-1)^j*A000032(n-(2*j+1)*k).

If (2*m+1)*k < n < (2*m+2)*k, and either k is odd and n+m is even, or k >= 4 is even and n is odd, then T(n,k) = Sum_{j=0..m} (-1)^j*A000032(n-(2*j+1)*k).

If (2*m+1)*k < n < (2*m+2)*k, and either k is odd and n+m is odd, or k >= 4 is even and n is even, then T(n,k) = -1 + Sum_{j=0..m} (-1)^j*A000032(n-(2*j+1)*k).

Example

Triangle starts:
   1;
   1,  1;
   2,  2,  1;
   3,  3,  1,  1;
   5,  5,  2,  1,  1;
   8,  8,  4,  2,  1,  1;
  13, 13,  6,  4,  2,  1,  1;
  21, 21, 10,  7,  4,  2,  1,  1;
  34, 34, 17, 11,  6,  4,  2,  1,  1;
  55, 55, 27, 18, 11,  6,  4,  2,  1,  1;
  89, 89, 44, 29, 17, 11,  6,  4,  2,  1,  1;

Maple

f:= (n, k) -> iquo(combinat:-fibonacci(n), combinat:-fibonacci(k)):
for n from 1 to 12 do
seq(f(n, k), k=1..n)
od:

Crossrefs

Cf. A000045, A000032, A357724.

Sequence in context: A372437 A144112 A178568 * A104660 A212125 A282936

Adjacent sequences: A357811 A357812 A357813 * A357815 A357816 A357817

Keyword

nonn,tabl,look

Author

J. M. Bergot and Robert Israel, Oct 13 2022

Status

approved