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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.

%I A368483 #38 Feb 02 2024 09:05:17
%S A368483 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,
%T A368483 67,73,79,84,91,98,105,112,119,126,135,143,151,159,168,177,187,196,
%U A368483 205,215,226,236,247,257,268,280,292,303,315,327,340,353,366
%N 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.
%C A368483 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) = ... .
%C A368483 Because of the Laurent property of a Somos-7 sequence the denominator of these terms is a monomial in the initial values.
%C A368483 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), ...
%C A368483 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.
%C A368483 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.
%C A368483 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.
%H A368483 S. Fomin and A. Zelevinsky, <a href="https://arxiv.org/abs/math/0104241">The Laurent phenomenon</a>, arXiv:math/0104241 [math.CO], 2001.
%H A368483 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,0,0,-1,-1,0,1).
%F A368483 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).
%F A368483 The growth rate is quadratic, a(n) = (7/60) * n^2 + O(n).
%o A368483 (Maxima) N : 7$ Len : 11$     /* Somos-N, N >= 2, Len = length of the calculated lists */
%o A368483 NofRT : floor (N / 2)$  /* number of terms in a Somos-N recurrence */
%o A368483 S : makelist (0, Len)$
%o A368483 G : makelist (0, Len)$ DegG : makelist (0, Len)$   /* G, the numerator of s() */
%o A368483 for i: 1 thru N do ( S[i] : s[i - 1], G[i] : 1, DegG[i] : 0 )$
%o A368483 for i: N + 1 thru Len do (
%o A368483    SS : 0,
%o A368483    for j : 1 thru NofRT do (
%o A368483       SS : SS + S[i - j] * S[i - N + j]
%o A368483    ),
%o A368483    S[i] : factor (SS / S[i - N]), G[i] : num (S[i]),
%o A368483    /* for N > 3 G is a homogenous polynomial, take the first monomial to determine the degree */
%o A368483    Mon : G[i], if N > 3 then ( Mon : args (Mon)[1] ),
%o A368483    DegG[i] : 0, for j : 0 thru N - 1 do ( DegG[i] : DegG[i] + hipow (Mon, s[j]) )
%o A368483 )$ DegG;
%Y A368483 Cf. A368052, A368481, A368482, A006723, A369611.
%K A368483 nonn,easy
%O A368483 0,8
%A A368483 _Helmut Ruhland_, Dec 26 2023