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Cardinality of the set f^n({s}), where f is a variant of the Collatz function that replaces any element x in the argument set with both x/2 and 3*x+1, and s is an arbitrary irrational number.
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%I #58 Apr 22 2023 10:27:25

%S 1,2,4,8,16,32,64,127,252,495,969,1886,3655,7054,13562,25978,49602,

%T 94440,179380,340001,643276,1215178,2292431,4319603,8131123,15292302,

%U 28738320,53970713,101297742,190028125,356319648,667866054,1251374689,2343968788,4389333758,8217535290,15381296139,28784811039,53859503664

%N Cardinality of the set f^n({s}), where f is a variant of the Collatz function that replaces any element x in the argument set with both x/2 and 3*x+1, and s is an arbitrary irrational number.

%C The start value can also be chosen as s = i, the imaginary unit.

%H Markus Sigg, <a href="/A208127/b208127.txt">Table of n, a(n) for n = 0..42</a>

%H Markus Sigg, <a href="/A208127/a208127_1.c.txt">C program to calculate as many terms as possible with given amount of memory</a>.

%F a(n) = |f^n({s})| where f(M) = {x/2 : x in M} union {3x+1 : x in M} and s is an arbitrary irrational number.

%e a(7) = 127 = 2^7-1 because there are exactly two 7-length sequences of h:=x->x/2 or t:=x->3*x+1 steps yielding the same value: (hhthtth)(s) = (thhhhtt)(s) = 27/16*s + 7/4. - _Alois P. Heinz_, Mar 30 2012

%p M := {sqrt(2)}:

%p print(nops(M)):

%p for i from 1 to 23 do

%p M := map(x -> x/2, M) union map(x -> 3*x+1, M):

%p print(nops(M))

%p end do:

%o (PARI) \\ maxGB is the available RAM memory size; use allocatemem() before start

%o a208127(maxGB) = {my (n=log(maxGB)/log(2)+21, v=[I]); for (i=0 , n, if(i>0, v=Set(concat(v/2,3*v+vector(#v,i,1)))); print1(#v,", "))};

%o a208127(16) \\ _Hugo Pfoertner_, Apr 09 2023

%K nonn

%O 0,2

%A _Markus Sigg_, Mar 29 2012

%E a(23)-a(25) from _Alois P. Heinz_, Mar 30 2012

%E a(26)-a(28) from _Markus Sigg_, Jul 05 2017

%E a(29)-a(31) from _Markus Sigg_, Aug 06 2017

%E a(32) from _Markus Sigg_, Mar 26 2023

%E a(33)-a(34) from _Hugo Pfoertner_, Mar 26 2023

%E a(35)-a(38) from _Markus Sigg_, Apr 06 2023