A129092 - OEIS

%I #11 Jan 12 2020 10:18:09

%S 1,2,5,16,69,430,4137,64436,1676353,74555322,5777029421,792086153688,

%T 194591768192733,86534148901444102,70244955881077121873,

%U 104827174339054175240700,289320796542222620694103961

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

%F 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.

%e The semi-Fibonacci sequence (A030067) starts:

%e [(1), 1, (2), 1, 3, 2, (5), 1, 6, 3, 9, 2, 11, 5, (16), 1, ...],

%e and obeys the recurrence:

%e A030067(n) = A030067(n/2) when n is even; and

%e A030067(n) = A030067(n-1) + A030067(n-2) when n is odd.

%e This sequence also equals row sums of triangle A129100:

%e     1;

%e     1,    1;

%e     2,    2,    1;

%e     5,    6,    4,   1;

%e    16,   24,   20,   8,   1;

%e    69,  136,  136,  72,  16,  1;

%e   430, 1162, 1360, 880, 272, 32, 1; ...

%e where columns of A129100 shift left under matrix square,

%e so that A129100^2 starts:

%e      1;

%e      2,    1;

%e      6,    4,   1;

%e     24,   20,   8,   1;

%e    136,  136,  72,  16,  1;

%e   1162, 1360, 880, 272, 32, 1; ...

%o (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])

%Y Cf. A030067, A129093, A129094.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 29 2007