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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified October 20 12:34 EDT 2018. Contains 316379 sequences. (Running on oeis4.)