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Meta-Fibonacci sequence a(n) with parameters s=2.
2

%I #13 Mar 16 2021 08:39:54

%S 1,1,1,2,2,2,2,3,4,4,4,4,4,5,6,6,7,8,8,8,8,8,8,9,10,10,11,12,12,12,13,

%T 14,14,15,16,16,16,16,16,16,16,17,18,18,19,20,20,20,21,22,22,23,24,24,

%U 24,24,25,26,26,27

%N Meta-Fibonacci sequence a(n) with parameters s=2.

%H C. Deugau and F. Ruskey, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Ruskey/ruskey6.pdf">Complete k-ary Trees and Generalized Meta-Fibonacci Sequences</a>, J. Integer Seq., Vol. 12. [This is a later version than that in the GenMetaFib.html link]

%H C. Deugau and F. Ruskey, <a href="http://www.cs.uvic.ca/~ruskey/Publications/MetaFib/GenMetaFib.html">Complete k-ary Trees and Generalized Meta-Fibonacci Sequences</a>

%H B. Jackson and F. Ruskey, <a href="https://doi.org/10.37236/1052">Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes</a>, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.

%F If 1 <= n <= 3, a(n)=1. If n = 4, then a(n)=2. If n>4 then a(n)=a(n-2-a(n-1)) + a(n-3-a(n-2)).

%F G.f.: A(z) = z * (1 - z^2) / (1 - z) * sum(prod(z^2 * (1 - z^(2 * [i])) / (1 - z^[i]), i=1..n), n=0..infinity), where [i] = (2^i - 1).

%p a := proc(n)

%p option remember;

%p if n <= 3 then return 1 end if;

%p if n <= 4 then return n-2 end if;

%p return add(a(n - i - 1 - a(n - i)), i = 1 .. 2)

%p end proc

%Y Cf. A120512, A120523.

%K nonn

%O 1,4

%A _Frank Ruskey_ and Chris Deugau (deugaucj(AT)uvic.ca), Jun 20 2006