I’ve been speculation on how the sequence behaves with large Seeds. Define the length of a run as the number of odd transitions (3n+1)/2 until a sequence is less than the Seed. The Seed 1_008_932_249_296_231 has a run with 559 odd steps. This 50 bit Seed has a run length over 11 times its width.
From trying runs in this region it seems to be the most efficient Seed. This can be attributed to a quirk of discrete arithmetic. With larger Seeds there are more bits in play and runs tend towards more average behavior due to entropy.
Even longer runs can be reached using Seeds that have many low order one bits. 1_008_932_249_296_231 has only 3 low order one bits. Seeds can be parameterized by the number of low order one bits and an even number that is any higher bits. For example, a Seed with 1000 low order one bits and upper bits 2380 has a run length of 2600. The performance ratio is then under 2.57; which is typical for this class of Seed.
I just have a simple PC and can’t cover many large runs. We’d get a better picture of long runs if someone that has a gaming rig with a few thousand cores could program it to cover larger Seeds.
Your seed is a “stopping time” (or “glide”) record according to the Table of Glide records from Eric Roosendaal. Its base-2 representation is:
11100101011001111001011000010100100101110101100111
The previous record in the table is 739448869367967, which in base-2 reads
10101000001000011001011011001100000011000010011111
It might be great if we could identify some sort of pattern in those binary words for all glide records, but nothing obvious apparently here, except the tendency to start with 3 or more bits set to one.
Not clear how you compute trajectories. I would suggest using Pari/GP for its multiprecision capabilities.
Thanks for the glide link. For the little computing I’ve done either Python or hand rolled multi-precision arithmetic is sufficient. I’ve found it most useful to define a length of a run as the number of odd steps until it goes below the seed. For the top few runs in the Glide table, counting the number of odds we get: