A228815 Symmetric triangle, read by rows, related to Fibonacci numbers.

0, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 10, 14, 10, 3, 5, 20, 36, 36, 20, 5, 8, 38, 83, 106, 83, 38, 8, 13, 71, 182, 281, 281, 182, 71, 13, 21, 130, 382, 690, 834, 690, 382, 130, 21, 34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34, 55, 420, 1546, 3586, 5780, 6750

Offset

0,5

Comments

Triangles satisfying the same recurrence: A091533, A091562, A185081, A205575, A209137, A209138.

Links

Table of n, a(n) for n=0..60.

Formula

G.f.: x*(1+y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.

Sum_{k = 0..n} T(n,k)*x^k = A000045(n), 2*A015518(n), 3*A015524(n), 4*A200069(n) for x = 0, 1, 2, 3 respectively.

Sum_{k = 0..floor(n/2)} T(n-k,k) = A008998(n+1).

Example

Triangle begins :
0
1, 1
1, 2, 1
2, 5, 5, 2
3, 10, 14, 10, 3
5, 20, 36, 36, 20, 5
8, 38, 83, 106, 83, 38, 8
13, 71, 182, 281, 281, 182, 71, 13
21, 130, 382, 690, 834, 690, 382, 130, 21
34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34
55, 420, 1546, 3586, 5780, 6750, 5780, 3586, 1546, 420, 55

Crossrefs

Cf. A000045 (1st column), A001629 (2nd column), A008998, A152011, A261055 (3rd column).

Sequence in context: A328601 A137327 A143913 * A241555 A277741 A241138

Adjacent sequences: A228812 A228813 A228814 * A228816 A228817 A228818

Keyword

nonn,easy,tabl

Author

Philippe Deléham, Oct 30 2013

Status

approved