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This sequence is the convolution of Fibonacci numbers with themselves.
The recurrence is
a[n] = a[n-1] + a[n-2] + Fibonacci[n+2]
If you assume Fibonacci sequence to start from 0,1,1,2,3,5 ... (http://oeis.org/A213587)
How can I generate it is logarithmic time or faster?
Please note that this is no homework nor any contest problem. I am working on Fibonacci applied sequences.
I was able to solve this in log n time by converting the recurrence relation to a fibonacci convolution..In the end ,the recurrence relation contained only Lucas Number and Fibonacci Number.So I was able to solve it in 2*log n .I will write the whole proof here once I figure out how to write mathematical symbols here.