Talk:Somos sequence

Ambiguous lead sentence

This strikes me as ambiguous:

It is not obvious from the form of their defining recurrence that every number in a Somos sequence is an integer, but nevertheless many Somos sequences have the property that all of their members are integers.

The phrase "It is not obvious from the form of their defining recurrence that every number in a Somos sequence is an integer" implies that every number in a Somos sequence is an integer. (As in, "It is not obvious, but it is the case that ...")

On the other hand, the phrase "but nevertheless many Somos sequences have the property that all of their members are integers" implies that some (and therefore not all) members of Somos sequences are integers.

So what is the truth here? Always integers? Sometimes integers, sometimes not? Or it's not certain?

Karl gregory jones (talk) 13:48, 31 August 2018 (UTC)

@Karl gregory jones: I agree. I have attempted to reword to make the situation clearer. (Incidentally, the lead section of the article is a brief summary of its contents -- there is a section of the body below titled "Integrality" that provides the full details.) --JBL (talk) 15:44, 31 August 2018 (UTC)

Somos-k for k >= 6 could have faster growth rates.

I think, the Somos-4 sequence and the Somos-5 sequence have 1 addition step, because ${\frac {a_{n-1}a_{n-3}a_{n-2}^{2}}{a_{n-4}}}$ and ${\frac {a_{n-1}a_{n-4}a_{n-2}a_{n-3}}{a_{n-5}}}$ would be just 1³ and 1^x is always 1.

But, if you do exactly 1 addition step, that is close to the end of the numerator, the growth rate becomes faster for larger k, because of more consecutive multiplications. The Somos-5 sequence has 4 factors, so it remains unchanged, when each product must be made from an even number of factors, which is why the Somos-2 sequence includes a_n-1² and the Somos-4 sequence includes a_n-2², but the Somos-6 sequence for example could be $a_{n}={\frac {a_{n-1}a_{n-5}a_{n-2}a_{n-4}+a_{n-3}^{2}}{a_{n-6}}}$.

rewritten Somos-6 sequence: $a_{n}={\frac {a_{n-1}a_{n-5}a_{n-2}a_{n-4}+a_{n-3}^{2}}{a_{n-6}}}$

1, 1, 1, 1, 1, 1, 2, 3, 7, 25, 359, 53899, 203172593, 638796642799202, 166404154799563946389343793, ...

rewritten Somos-7 sequence: $a_{n}={\frac {a_{n-1}a_{n-6}a_{n-2}a_{n-5}+a_{n-3}a_{n-4}}{a_{n-7}}}$

1, 1, 1, 1, 1, 1, 1, 2, 3, 7, 23, 167, 7703, 7718567, 624289278731, 258599199555990647066, 88584822649895547931920734367705641

rewritten Somos-8 sequence: $a_{n}={\frac {a_{n-1}a_{n-7}a_{n-2}a_{n-6}a_{n-3}a_{n-5}+a_{n-4}^{2}}{a_{n-8}}}$

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 7, 43, 907, 546023, 127772658187, 2657696865520408028143, 83716693232323891118566007843531807067239, 2587068736119503100677143091363581874356540199431912907761079123903282521500074, 1751112777024626062964949248705121935518944298337156370379574647642301955440689021918817717363414396287751054936420534709674818155939480270309340550889 94.31.88.138 (talk) 18:16, 18 September 2023 (UTC)

This page is for discussing improvements to the article based on reliable, published sources, not for conducting original research related to the topic of the article. --JBL (talk) 22:44, 18 September 2023 (UTC)