Sequence-Counting Sequences: Variants of A008934

[Paul D. Hanna]

Seqfans,
         Thanks, Brendan, for the comments regarding the recurrence. 
I have found 3 distinct recurrences, and I am working on one more. 
There are 2 surprising recurrences involving powers of matrices, and 
also I found the generalization of the Cook-Kleber recurrence for all
m>1.
 
I am preparing this whole family of sequences for submission to OEIS 
(it may take me up to a week before they are finalized and submitted).
 
Question: since these sequences are a natural extension of A008934, 
which is the number of tournament sequences (where m=2), 
what name could I assign these related sequences? 
Should I refer to them as "k-tournament sequences" - any suggestions? 
  
I am referring to the sequence-counting sequences defined by: 
"Number of sequences (a_1, a_2,..., a_n) with a_1 = 1 
such that a_i < a_{i+1} <= m*a_i for all i." where m>1.
 
For example, when m=3 we obtain these terms using the recurrences:
1,2,10,114,2970,182402,27392682,10390564242,10210795262650,
26494519967902114,184142934938620227530,
3466516611360924222460082,178346559667060145108789818842,
25264074391478558474014952210052802 
which exactly agrees with explicit counting (that obtained only the
initial 8 terms).
 
Thanks, 
      Paul
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