A129093 a(n) = A030067(2^(n+1) - 3) for n>=1, where A030067 is the semi-Fibonacci numbers.

1, 3, 11, 53, 361, 3707, 60299, 1611917, 72878969, 5702474099, 786309124267, 193799682039045, 86339557133251369, 70158421732175677771, 104756929383173098118827, 289215969367883566518863261

Offset

1,2

Links

Table of n, a(n) for n=1..16.

Formula

Equals the first differences of A129092: a(n) = A129092(n+1) - A129092(n).

Equals the row sums of the matrix square of triangle A129100.

Example

This sequence also equals the row sums of the triangle formed from the semi-Fibonacci numbers (A030067) with 2^n terms in row n for n>=0:
n=0: 1;
n=1: 1, 2;
n=2: 1, 3, 2, 5;
n=3: 1, 6, 3, 9, 2, 11, 5, 16;
n=4: 1, 17, 6, 23, 3, 26, 9, 35, 2, 37, 11, 48, 5, 53, 16, 69; ...
and the rightmost border equals A129092(n) = A030067(2^n - 1).
The semi-Fibonacci numbers (A030067) start:
[1, (1), 2, 1, (3), 2, 5, 1, 6, 3, 9, 2, (11), 5, 16, 1, ...],
and obey the recurrence:
A030067(n) = A030067(n/2) when n is even; and
A030067(n) = A030067(n-1) + A030067(n-2) when n is odd.
This sequence also equals row sums of matrix square A129100^2:
1;
2, 1;
6, 4, 1;
24, 20, 8, 1;
136, 136, 72, 16, 1;
1162, 1360, 880, 272, 32, 1; ...

Prog

(PARI) /* As row sums of the matrix square of triangle A129100: */
a(n)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(r=1, m, for(c=1, r, if(r==c || r==1 || r==2, B[r, c]=1, if(c==1, B[r, 1]=sum(i=1, r-1, A[r-1, i]), B[r, c]=(A^(2^(c-1)))[r-c+1, 1])))); A=B); sum(k=1, n, (A^2)[n, k])
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A030067, A129092, A129094.

Sequence in context: A224345 A081367 A156171 * A259106 A057325 A302760

Adjacent sequences: A129090 A129091 A129092 * A129094 A129095 A129096

Keyword

nonn

Author

Paul D. Hanna, Mar 29 2007

Status

approved