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Generalized meta-Fibonacci sequence a(n) with parameters s=1 and k=4.
2

%I #9 Mar 16 2021 05:28:58

%S 1,1,2,3,4,4,4,5,6,7,8,8,9,10,11,12,12,13,14,15,16,16,16,16,17,18,19,

%T 20,20,21,22,23,24,24,25,26,27,28,28,29,30,31,32,32,32,33,34,35,36,36,

%U 37,38,39,40,40,41,42,43,44,44

%N Generalized meta-Fibonacci sequence a(n) with parameters s=1 and k=4.

%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>

%F If 1 <= n <= 2, a(n)=1. If 3 <= n <= 5, then a(n)=n-1. If n>5 then a(n)=a(n-1-a(n-1)) + a(n-2-a(n-2)) + a(n-3-a(n-3)) + a(n-4-a(n-4)).

%F G.f.: A(z) = z * sum(prod(z * (1 - z^(4 * [i])) / (1 - z^[i]), i=1..n), n=0..infinity), where [i] = (4^i - 1) / 3.

%p a := proc(n)

%p option remember;

%p if n <= 2 then return 1 end if;

%p if n <= 5 then return n-1 end if;

%p return add(a(n - i - a(n - i)), i = 1 .. 4)

%p end proc

%Y Cf. A120519, A120530.

%K nonn

%O 1,3

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