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

%I A129092 #11 Jan 12 2020 10:18:09
%S A129092 1,2,5,16,69,430,4137,64436,1676353,74555322,5777029421,792086153688,
%T A129092 194591768192733,86534148901444102,70244955881077121873,
%U A129092 104827174339054175240700,289320796542222620694103961
%N A129092 a(n) = A030067(2^n - 1) for n >= 1, where A030067 is the semi-Fibonacci numbers.
%F A129092 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 A129092 The semi-Fibonacci sequence (A030067) starts:
%e A129092 [(1), 1, (2), 1, 3, 2, (5), 1, 6, 3, 9, 2, 11, 5, (16), 1, ...],
%e A129092 and obeys the recurrence:
%e A129092 A030067(n) = A030067(n/2) when n is even; and
%e A129092 A030067(n) = A030067(n-1) + A030067(n-2) when n is odd.
%e A129092 This sequence also equals row sums of triangle A129100:
%e A129092     1;
%e A129092     1,    1;
%e A129092     2,    2,    1;
%e A129092     5,    6,    4,   1;
%e A129092    16,   24,   20,   8,   1;
%e A129092    69,  136,  136,  72,  16,  1;
%e A129092   430, 1162, 1360, 880, 272, 32, 1; ...
%e A129092 where columns of A129100 shift left under matrix square,
%e A129092 so that A129100^2 starts:
%e A129092      1;
%e A129092      2,    1;
%e A129092      6,    4,   1;
%e A129092     24,   20,   8,   1;
%e A129092    136,  136,  72,  16,  1;
%e A129092   1162, 1360, 880, 272, 32, 1; ...
%o A129092 (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 A129092 Cf. A030067, A129093, A129094.
%K A129092 nonn
%O A129092 1,2
%A A129092 _Paul D. Hanna_, Mar 29 2007