A129092 a(n) = A030067(2^n - 1) for n >= 1, where A030067 is the semi-Fibonacci numbers.

1, 2, 5, 16, 69, 430, 4137, 64436, 1676353, 74555322, 5777029421, 792086153688, 194591768192733, 86534148901444102, 70244955881077121873, 104827174339054175240700, 289320796542222620694103961

Offset

1,2

Links

Table of n, a(n) for n=1..17.

Formula

Equals the row sums and first column of triangle A129100: a(n) = A129100(n,0), where column 0 of matrix power A129100^(2^k) = column k of A129100 for k > 0.

Example

The semi-Fibonacci sequence (A030067) starts:
[(1), 1, (2), 1, 3, 2, (5), 1, 6, 3, 9, 2, 11, 5, (16), 1, ...],
and obeys 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 triangle A129100:
    1;
    1,    1;
    2,    2,    1;
    5,    6,    4,   1;
   16,   24,   20,   8,   1;
   69,  136,  136,  72,  16,  1;
  430, 1162, 1360, 880, 272, 32, 1; ...
where columns of A129100 shift left under matrix square,
so that A129100^2 starts:
     1;
     2,    1;
     6,    4,   1;
    24,   20,   8,   1;
   136,  136,  72,  16,  1;
  1162, 1360, 880, 272, 32, 1; ...

Prog

(PARI) /* Generated as column 0 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); return(A[n+1, 1])

Crossrefs

Cf. A030067, A129093, A129094.

Sequence in context: A059295 A259408 A251684 * A110710 A245881 A078639

Adjacent sequences: A129089 A129090 A129091 * A129093 A129094 A129095

Keyword

nonn

Author

Paul D. Hanna, Mar 29 2007

Status

approved