A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)*T(n-1,k) + 1]/T(n-2,k-1) for 0<k<n with T(n,n) = 1 for n>=0 and T(n,0) = Fibonacci(2*n-1) for n>=1.

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 13, 13, 8, 4, 1, 34, 34, 21, 11, 5, 1, 89, 89, 55, 29, 14, 6, 1, 233, 233, 144, 76, 37, 17, 7, 1, 610, 610, 377, 199, 97, 45, 20, 8, 1, 1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1, 4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1080, as a flattened table of rows 0..45

Formula

T(n,k) = Fibonacci(2*(n-k)-1) + k*Fibonacci(2*(n-k)) for 0<=k<=n.

Example

Generating rule.
Given nonzero elements W, X, Y, Z, relatively arranged like so:
.. W .....
.. X Y ...
.... Z ...
then Z = (X*Y + 1)/W.
Triangle begins:
1;
1, 1;
2, 2, 1;
5, 5, 3, 1;
13, 13, 8, 4, 1;
34, 34, 21, 11, 5, 1;
89, 89, 55, 29, 14, 6, 1;
233, 233, 144, 76, 37, 17, 7, 1;
610, 610, 377, 199, 97, 45, 20, 8, 1;
1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1;
4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1; ...

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, if(k==0, fibonacci(2*n-1), (T(n-1, k-1)*T(n-1, k)+1)/T(n-2, k-1))))
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) T(n, k)=fibonacci(2*(n-k))*k+fibonacci(2*(n-k)-1)
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. row sums: A141752; columns: A001519, A001906, A002878, A054486, A054492, A055267, A055273, A055849.

Sequence in context: A299499 A190215 A190252 * A079222 A033184 A171567

Adjacent sequences: A141748 A141749 A141750 * A141752 A141753 A141754

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 04 2008

Status

approved