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

0,2

Comments

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

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.

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.

Example

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

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:

Prog

(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

Crossrefs

Sequence in context: A335247 A145113 A062257 * A172316 A062258 A239560

Adjacent sequences: A208124 A208125 A208126 * A208128 A208129 A208130

Keyword

nonn

Author

Markus Sigg, Mar 29 2012

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

Status

approved