A094570 Triangle T(n,k) read by rows: T(n,k) = F(k) + F(n-k) where F(n) is the n-th Fibonacci number.

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 5, 4, 3, 3, 4, 5, 8, 6, 4, 4, 4, 6, 8, 13, 9, 6, 5, 5, 6, 9, 13, 21, 14, 9, 7, 6, 7, 9, 14, 21, 34, 22, 14, 10, 8, 8, 10, 14, 22, 34, 55, 35, 22, 15, 11, 10, 11, 15, 22, 35, 55, 89, 56, 35, 23, 16, 13, 13, 16, 23, 35, 56, 89, 144, 90, 56, 36, 24, 18, 16, 18, 24, 36, 56, 90, 144

Offset

0,5

Links

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened

Formula

Row n: F(0)+F(n), F(1)+F(n-1), F(2)+F(n-2), ..., F(n-1)+F(1), F(n)+F(0).

From Reinhard Zumkeller, Mar 21 2011: (Start)

T(n,0) = T(n,n) = A000045(n).

T(2*n,n) = A006355(n+1).

T(n,k) = A141169(n,k) + A141169(n,n-k). (End)

Sum(T(n,k), 0<=k<=n) = 2*A000071(n+2) = 2*A000045(n+2) - 2. - Philippe Deléham, Apr 07 2013

Example

Triangle begins:
0;
1, 1;
1, 2, 1;
2, 2, 2, 2;
3, 3, 2, 3, 3;
5, 4, 3, 3, 4, 5;
8, 6, 4, 4, 4, 6, 8;
13, 9, 6, 5, 5, 6, 9, 13;
21, 14, 9, 7, 6, 7, 9, 14, 21;

Prog

(PARI) row(n) = vector(n+1, k, k--; fibonacci(k)+fibonacci(n-k)); \\ Michel Marcus, Mar 22 2021

Crossrefs

Cf. A000045, A000071, A006355, A141169.

Sequence in context: A303399 A053597 A230197 * A225638 A332656 A230443

Adjacent sequences: A094567 A094568 A094569 * A094571 A094572 A094573

Keyword

nonn,tabl

Author

Clark Kimberling, May 12 2004

Status

approved