OFFSET
1,3
COMMENTS
a(n+1) is the number of multus bitstrings of length n with no runs of 8 ones. - Steven Finch, Mar 25 2020
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..200
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
Robert Israel, Proof of empirical recurrence
FORMULA
Empirical: a(n) = a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) + a(n-8).
Empirical g.f.: -x*(1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7) / ( (x^2 + 1)*(x^6 + x^5 + x^2 + x - 1) ). - R. J. Mathar, Jul 25 2012
Empirical recurrence verified (see link). - Robert Israel, Jan 27 2019
EXAMPLE
Some solutions for n=13:
1 1 1 4 1 5 1 6 1 1 1 5 3 3 5 1
8 2 2 1 2 1 2 1 2 2 4 1 1 1 1 4
2 3 3 2 3 2 3 2 10 3 2 2 2 2 2 2
3 4 4 3 7 3 4 3 3 4 3 3 4 8 3 3
4 5 7 9 4 4 7 4 4 12 5 4 5 4 4 5
5 8 5 5 5 6 5 5 5 5 6 6 11 5 9 10
6 6 6 6 6 9 6 7 6 6 7 12 6 6 6 6
7 7 12 7 8 7 8 8 7 7 8 7 7 7 7 7
13 13 8 8 9 8 13 9 8 8 9 8 8 13 8 8
9 9 9 10 12 10 9 10 9 9 10 9 9 9 10 9
10 10 10 11 10 13 10 11 11 10 13 10 10 10 13 11
11 11 11 12 11 11 11 12 12 11 11 11 12 11 11 12
12 12 13 13 13 12 12 13 13 13 12 13 13 12 12 13
MAPLE
f:= proc(n) option remember; local k;
if n < 0 then return 0 fi;
f(n-1) + add(f(n-k), k=3..8)
end proc:
f(0):= 1:
map(f, [$1..60]); # Robert Israel, Jan 27 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Apr 24 2011
STATUS
approved