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%I #14 May 16 2020 03:48:02
%S 1,2,3,5,8,13,21,33,52,81,126,194,296,451,682,1031,1548,2314,3453,
%T 5125,7593,11212,16508,24250,35525,51931,75734,110239,160135,232178,
%U 336041,485529,700401,1008750,1450770,2083412
%N Iterate function composition by applying y=abs(x) or y=x-1. a(n) is the number of functions at distance n from the identity (y=x) in the graph of all possible results.
%C The first 7 terms of this sequence are Fibonacci numbers. There indeed is an imperfect analogy with the rabbit problem: view applying y=x-1 as aging 1 month and applying y=abs(x) as [having a good chance of] procreating/being born; then the fact that the nodes of the abs(something) form do not generate new nodes when y=abs(x) is applied to them can be viewed as first month's immaturity. The reason why this sequence deviates from Fibonacci's is the existence of identities such as abs(abs(abs(abs(x)-1)-1)-1) = abs(abs(abs(x)-2)-1), and this precise example proves that abs(abs(abs(x)-1)-1)-1, "although aged >=1, is immature".
%H Luc Rousseau, <a href="/A334738/a334738.cat.txt">A Java program that computes A334738.</a>
%e 0: 1: 2: 3: 4: 5:
%e -----------------------------------------------------------------------
%e x x-1 x-2 x-3 x-4 x-5
%e -----------
%e |x-4|
%e -----------------------
%e |x-3| |x-3|-1
%e -----------------------------------
%e |x-2| |x-2|-1 |x-2|-2
%e -----------
%e ||x-2|-1|
%e -----------------------------------------------
%e |x-1| |x-1|-1 |x-1|-2 |x-1|-3
%e -----------
%e ||x-1|-2|
%e -----------------------
%e ||x-1|-1| ||x-1|-1|-1
%e -----------------------------------------------------------
%e |x| |x|-1 |x|-2 |x|-3 |x|-4
%e -----------
%e ||x|-3|
%e -----------------------
%e ||x|-2| ||x|-2|-1
%e -----------------------------------
%e ||x|-1| ||x|-1|-1 ||x|-1|-2
%e -----------
%e |||x|-1|-1|
%e -----------------------------------------------------------------------
%o (Java) see link.
%Y Cf. A000045.
%K nonn,more
%O 0,2
%A _Luc Rousseau_, May 09 2020
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