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A B-tree of order m is an ordered tree such that every node has at most m children, the root has at least 2 children, every node except for the root has 0 or at least m/2 children, all end-nodes are at the same level.

Lim_{n->infinity} a(n)^(1/n) = (1+sqrt(5))/2, for more detailed asymptotics see Odlyzko 1982 reference. - Vaclav Kotesovec, Jul 29 2014

From Jianing Song, Nov 02 2019: (Start)

For n > 0, a(n) is also number of length-n sequences (d_1, d_2, ..., d_n) such that: (a) d_1 = 0, d_i > 0 for 2 <= i <= n; (b) for all 1 <= t <= n, at least one of d_i and d_(i+1) is equal to M = max_{t=1..n} d_t; (c) for all 1 <= i < j <= n+1, if max{d_i, d_j} < d_t for i < t < j, then between d_i and d_j there are exactly 1 or 2 terms equal to max{d_i, d_j} + 1. Here d_(n+1) = d_1. For example, for n = 8 there are four such sequences: (0, 3, 2, 3, 1, 3, 2, 3), (0, 2, 2, 1, 2, 2, 1, 2), (0, 2, 2, 1, 2, 1, 2, 2), (0, 2, 1, 2, 2, 1, 2, 2). For convention let's call such sequences "R sequences with largest term M".

Note that for M > 0, (0, d_2, d_3, ..., d_n1, 1, e_2, e_3, ..., e_n2) (d_t, e_u > 1) is an "R sequence with largest term M" if and only if (0, d_2-1, d_3-1, ..., d_n1-1) and (0, e_2-1, e_3-1, ..., e_n2-1) are both "R sequences with largest term M-1"; similarly, (0, d_2, d_3, ..., d_n1, 1, e_2, e_3, ..., e_n2, 1, f_2, f_3, ..., f_n3) (d_t, e_u, f_v > 1) is an "R sequence with largest term M" if and only if (0, d_2-1, d_3-1, ..., d_n1-1), (0, e_2-1, e_3-1, ..., e_n2-1) and (0, f_2-1, f_3-1, ..., f_n3-1) are all "R sequences with largest term M-1". From this we can see that each "R sequence with largest term M" of length-n is isomorphic to a B-tree of order 3 with M levels and n leaves, where the root is counted as the 0th level.

The condition (c) above is equivalent to: (c') there are no three or more consecutive M's in the sequence; if we eliminate all the M's, we get a shorter "R sequence with largest term M-1".

The number of B-trees of order 3 with M levels or the number of "R sequences with largest term M" is given by A125295(M). (End)

a(7) has 266 decimal digits and a(8) has 796 decimal digits. - Andrew Howroyd, Sep 10 2019

Also the number of chordless cycles in the (n+1)-Hanoi graph. - Eric W. Weisstein, May 23 2023