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

%I A129093 #10 Jun 14 2017 00:26:40
%S A129093 1,3,11,53,361,3707,60299,1611917,72878969,5702474099,786309124267,
%T A129093 193799682039045,86339557133251369,70158421732175677771,
%U A129093 104756929383173098118827,289215969367883566518863261
%N A129093 a(n) = A030067(2^(n+1) - 3) for n>=1, where A030067 is the semi-Fibonacci numbers.
%F A129093 Equals the first differences of A129092: a(n) = A129092(n+1) - A129092(n).
%F A129093 Equals the row sums of the matrix square of triangle A129100.
%e A129093 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:
%e A129093 n=0: 1;
%e A129093 n=1: 1, 2;
%e A129093 n=2: 1, 3, 2, 5;
%e A129093 n=3: 1, 6, 3, 9, 2, 11, 5, 16;
%e A129093 n=4: 1, 17, 6, 23, 3, 26, 9, 35, 2, 37, 11, 48, 5, 53, 16, 69; ...
%e A129093 and the rightmost border equals A129092(n) = A030067(2^n - 1).
%e A129093 The semi-Fibonacci numbers (A030067) start:
%e A129093 [1, (1), 2, 1, (3), 2, 5, 1, 6, 3, 9, 2, (11), 5, 16, 1, ...],
%e A129093 and obey the recurrence:
%e A129093 A030067(n) = A030067(n/2) when n is even; and
%e A129093 A030067(n) = A030067(n-1) + A030067(n-2) when n is odd.
%e A129093 This sequence also equals row sums of matrix square A129100^2:
%e A129093 1;
%e A129093 2, 1;
%e A129093 6, 4, 1;
%e A129093 24, 20, 8, 1;
%e A129093 136, 136, 72, 16, 1;
%e A129093 1162, 1360, 880, 272, 32, 1; ...
%o A129093 (PARI) /* As row sums of the matrix square of triangle A129100: */
%o A129093 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])
%o A129093 for(n=1,20,print1(a(n),", "))
%Y A129093 Cf. A030067, A129092, A129094.
%K A129093 nonn
%O A129093 1,2
%A A129093 _Paul D. Hanna_, Mar 29 2007