1,2

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.

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 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; ...

(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])}

Cf. A030067; A129093, A129094.

Adjacent sequences: A129089 A129090 A129091 this_sequence A129093 A129094 A129095

Sequence in context: A107948 A058673 A059295 this_sequence A110710 A002632 A020127

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 29 2007