

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.


3



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
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OFFSET

0,2


COMMENTS

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


LINKS



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 = 2^71 because there are exactly two 7length 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, ", "))};


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



