Somos Sequences

I came across an interesting article in Quanta Magazine, dated November 16th 2023, that mentions Somos sequences that I'd not heard of before. To quote from the article:

A Somos-\(k\) sequence starts with the digit or digits 1, \(k\) of them. Each new term of a Somos-\(k\) sequence is defined by pairing off previous terms, multiplying each pair together, adding up the pairs, and then dividing by the term \(k\) positions back in the sequence.

The sequences aren’t very interesting if \(k\) equals 1, 2 or 3 — they are just a series of repeating ones. But for \(k\) = 4, 5, 6 or 7 the sequences have a weird property. Even though there is a lot of division involved, fractions don’t appear.

Figure 1 shows an excerpt from the article that illustrates how the Somos-5 series is generated.

Figure 1

It's easy enough to write some SageMath code to generate the Somos-5 and here are the initial terms and, as can be seen, the terms get very large very quickly.

1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933, 11548470571, 142844426789, 2279343327171, 57760865728994, 979023970244321

These terms constitute OEIS A006721  (permalink): 

A006721

Somos-5 sequence: a(n) = (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5), with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

The Somos-4 sequence is OEIS A006720 (permalink):

A006720

Somos-4 sequence: a(0)=a(1)=a(2)=a(3)=1; for n >= 4, a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).

The initial members are:

1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786, 1662315215971057, 61958046554226593, 4257998884448335457, 334806306946199122193

The Somos-6 series is OEIS A006722:

A006722

Somos-6 sequence: a(n) = (a(n-1) * a(n-5) + a(n-2) * a(n-4) + a(n-3)^2) / a(n-6), a(0) = ... = a(5) = 1.

The initial members are:

1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875, 45788778247, 805144998681, 14980361322965, 620933643034787, 16379818848380849, 369622905371172929, 20278641689337631649, 995586066665500470689

The Somos-7 series is OEIS A006723:

A006723

Somos-7 sequence: a(n) = (a(n-1) * a(n-6) + a(n-2) * a(n-5) + a(n-3) * a(n-4)) / a(n-7), a(0) = ... = a(6) = 1.

The initial members are:

1, 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, 1925, 7203, 34081, 227321, 1737001, 14736001, 63232441, 702617001, 8873580481, 122337693603, 1705473647525, 22511386506929, 251582370867257, 9254211194697641, 215321535159114017