A208127 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.
1, 2, 4, 8, 16, 32, 64, 127, 252, 495, 969, 1886, 3655, 7054, 13562, 25978, 49602, 94440, 179380, 340001, 643276, 1215178, 2292431, 4319603, 8131123, 15292302, 28738320, 53970713, 101297742, 190028125, 356319648, 667866054, 1251374689, 2343968788, 4389333758, 8217535290, 15381296139, 28784811039, 53859503664
Offset: 0
Keywords
Examples
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
Links
- Markus Sigg, Table of n, a(n) for n = 0..42
- Markus Sigg, C program to calculate as many terms as possible with given amount of memory.
Programs
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Maple
M := {sqrt(2)}: print(nops(M)): for i from 1 to 23 do M := map(x -> x/2, M) union map(x -> 3*x+1, M): print(nops(M)) end do: -
PARI
\\ maxGB is the available RAM memory size; use allocatemem() before start 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,", "))}; a208127(16) \\ Hugo Pfoertner, Apr 09 2023
Formula
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.
Extensions
a(23)-a(25) from Alois P. Heinz, Mar 30 2012
a(26)-a(28) from Markus Sigg, Jul 05 2017
a(29)-a(31) from Markus Sigg, Aug 06 2017
a(32) from Markus Sigg, Mar 26 2023
a(33)-a(34) from Hugo Pfoertner, Mar 26 2023
a(35)-a(38) from Markus Sigg, Apr 06 2023
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