Hello, In A001006 there is the recursion (n+2)a(n)=(2n+1)a(n-1)+(3n-3)a(n-2) Using it with Maple, I got the first 1000 terms in a fraction of a second. It occurs, with a hint for the proof, in E. Barcucci, R. Pinzani, R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279. which will be added to A001006. I will ask Neil to add also lim(a(n)/a(n-1), n->infinity) = 3 (from the Aigner paper) and a(n) ~ 3^(n+1)sqrt(3)[1+1/(16n)]/[(2n+3)sqrt((n+2)Pi)] from the Barcucci et al. paper Emeric Deutsch On Tue, 23 Mar 2004, Simon Plouffe wrote:
hello,
(this is old material from 1993),
I collected some of my notes about exact formulas for integer sequences. They are somewhat usefull since it can be used to compute the n'th term of a sequence.
it uses [ ] , { } and some constants like exp(1).
http://www.lacim.uqam.ca/%7Eplouffe/exact.htm
note : most of those formulas do not appear in the current On-line Encyclopedia of integer sequences.
By the way, can anybody do this : Can we find an EXACT formula for the n'th term of A001006 (Motzkin numbers), the only useful formula we have is a recurrence but with the index that grows with n which makes it difficult to compute for large n. http://www.research.att.com/projects/OEIS?Anum=A001006 is 1,1,2,4,9,21,51,127,323,835,2188,5798, ... and it grows roughly like 3^n.
Simon Plouffe
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