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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A368483 The degree of polynomials related to Somos-7 sequences. Also for n > 6 the degree of the (n-6)-th involution in a family of involutions in the Cremona group of rank 6 defined by a Somos-7 sequence.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 7, 9, 12, 14, 16, 19, 23, 26, 30, 33, 37, 42, 47, 51, 56, 61, 67, 73, 79, 84, 91, 98, 105, 112, 119, 126, 135, 143, 151, 159, 168, 177, 187, 196, 205, 215, 226, 236, 247, 257, 268, 280, 292, 303, 315, 327, 340, 353, 366
Offset: 0

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Author

Helmut Ruhland, Dec 26 2023

Keywords

Comments

Let s(0), s(1), ..., s(5), s(6) be the 7 initial values in a Somos-7 sequence. The following terms s(7), s(8), ... are rational expressions in the 7 initial values derived from the Somos-7 recurrence: s(n) = ( s(n-1)*s(n-6) + s(n-2)*s(n-5) + s(n-3)*s(n-4) ) / s(n-7). E.g., s(7) = (s(1)*s(6) + s(2)*s(5) + s(3)*s(4)) / s(0), s(8) = ... .
Because of the Laurent property of a Somos-7 sequence the denominator of these terms is a monomial in the initial values.
With the sequence e(n) = A369611(n), the tropical version of the Somos-7 sequence, the monomial D(n) is defined as Product_{k=0..6} s(k)^a(n-k). Define the polynomial G(n) to be s(n) * D(n). G(n) is 1 for n < 7, else G(n) is the numerator of s(n), so ..., G(5) = 1, G(6) = 1, G(7) = s(1)*s(6) + s(2)*s(5) + s(3)*s(4), ...
For n >= 0, a term a(n) of the actual sequence is the degree of G(n). The degree of the denominator of s(n) is a(n) - 1.
This Somos-7 sequence defines a family (proposed name: Somos family) S of (birational) involutions in Cr_6(R), the Cremona group of rank 6.
A Somos involution S(n) in this family is defined as S(n) : RP^6 -> RP^6, (s(0) : s(1) : ... : s(5) : s(6)) -> (s(n+6) : s(n+5) : ... : s(n+1) : s(n)). For n > 0 S(n) = (G(n+6) : G(n+5)*m1 : ... : G(n+1)*m5 : G(n)*m6 ), with m1, m2, ..., m5, m6 monomials. The involutions generate an infinite dihedral group. Already 2 consecutive involutions S(n), S(n+1) generate this group too. This group as a dihedral group has 2 conjugacy classes { ..., S(0), S(2), S(4), ... } and { ..., S(1), S(3), S(5), ... } of involutions. The degree of such an involution S(n) equals the degree of G(n+6) and the term a(n+6) in the actual sequence.

Crossrefs

Programs

  • Maxima
    N : 7$ Len : 11$     /* Somos-N, N >= 2, Len = length of the calculated lists */
    NofRT : floor (N / 2)$  /* number of terms in a Somos-N recurrence */
    S : makelist (0, Len)$
    G : makelist (0, Len)$ DegG : makelist (0, Len)$   /* G, the numerator of s() */
    for i: 1 thru N do ( S[i] : s[i - 1], G[i] : 1, DegG[i] : 0 )$
    for i: N + 1 thru Len do (
       SS : 0,
       for j : 1 thru NofRT do (
          SS : SS + S[i - j] * S[i - N + j]
       ),
       S[i] : factor (SS / S[i - N]), G[i] : num (S[i]),
       /* for N > 3 G is a homogenous polynomial, take the first monomial to determine the degree */
       Mon : G[i], if N > 3 then ( Mon : args (Mon)[1] ),
       DegG[i] : 0, for j : 0 thru N - 1 do ( DegG[i] : DegG[i] + hipow (Mon, s[j]) )
    )$ DegG;

Formula

a(n) = 1 + e(n-6) + e(n-5) + e(n-4) + e(n-3) + e(n-2) + e(n-1) + e(n), where e(n) = A369611(n), the tropical version of Somos-7, is the exponent of one of the initial values in the denominator of s(n).
The growth rate is quadratic, a(n) = (7/60) * n^2 + O(n).
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