OFFSET
1,1
COMMENTS
Sequence A198584 gives the first term of the Collatz sequence having exactly 3 odd numbers. This sequence is the subset of A198584 for which the second odd number is larger than the first. The second odd number is (2^(6*n - 2) - 1)/3, which always occurs as the third term of the sequence.
{a(n) mod 6} = {repeat(3, 5, 1)}, and a(n) mod 8 = 3 for all n. Proof from the formula of a(n) in terms of A198586 given below, using the modulo 72 congruence of the odd indexed part of A198586 given there. - Wolfdieter Lang, Jan 14 2022
LINKS
FORMULA
a(n) = (64^n/2 - 5)/9. - Alois P. Heinz, Dec 08 2021
From Wolfdieter Lang, Jan 12 2022: (Start)
a(n) = (2*A198586(2*n-1) - 1)/3. See the Mathematica program.
G.f.: x*(3 + 32*x)/((1 - x)*(1 - 64*x)). (End)
EXAMPLE
The number 3 has the Collatz iteration {3, 10, 5, 16, 8, 4, 2, 1}, which has three out-of-order odd numbers {3, 5, 1}.
MATHEMATICA
Table[(2*(2^(6*n - 2) - 1)/3 - 1)/3, {n, 15}]
PROG
(PARI) a(n)=4^(3*n-1)\3*2\3 \\ Charles R Greathouse IV, Mar 11 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
T. D. Noe, Sep 12 2013
STATUS
approved