Conjecture commented on A144841 wrong?

[Taichi Aoki]

Hi,
I see this comment on A144841

"Conjectured values for maximal number of regions obtained by joining each triple of n points on a 3 dimensional sphere by a plane."
However, my math friends in Japan was saying that their should be 48 regions for n=5 instead of the 41 written in the sequence. (https://x.com/sugaku_day/status/1988617807647391992) 

I was able to visualize it. (n=5 green points, and 48 regions with purple points inside)

https://www.desmos.com/3d/2u3gnbfcda

Does this mean that we found a counter example to the conjecture, or am I just misinterpreting the comment? 

[Seiichi Manyama]

Hi,

Thank you for your email.

The problem of partitioning a sphere by binomial(n,3) planes determined by n points is discussed in a thread on Math StackExchange:
https://math.stackexchange.com/questions/1998893/maximum-number-of-regions-of-a-sphere-partitioned-by-binomn3-planes-from?utm_source=chatgpt.com

It appears quite plausible that the value f(5)=41 currently listed (e.g. in the OEIS entry for A144841) may indeed be too low.

Best regards,
Seiichi

2025年11月13日(木) 17:15 Seiichi Manyama <manc...@gmail.com>:

Hi,

Thank you for your email.

I wanted to point out that, in fact, the problem of partitioning a sphere by (3n) planes determined by n points is discussed in a thread on Math StackExchange:
https://math.stackexchange.com/questions/1998893/maximum-number-of-regions-of-a-sphere-partitioned-by-binomn3-planes-from?utm_source=chatgpt.com

It appears quite plausible that the value f(5)=41 currently listed (e.g. in the OEIS entry for A144841) may indeed be too low.

Best regards,
Seiichi

2025年11月13日(木) 2:00 Taichi Aoki <aoki198...@gmail.com>:

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[Taichi Aoki]

Dear Seiichi-san,

Thank you for the reply!
I will ask the original author where that conjecture came from.

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAE4WgJGGtto%2B-W2M1DE-XWbgnVu91geN7XAinj_OQsKcb7xXLA%40mail.gmail.com.