%I #25 Feb 02 2024 07:47:21

%S 0,0,0,0,0,0,2,3,4,6,8,11,13,16,20,23,27,31,36,41,45,51,57,63,69,75,

%T 83,90,97,105,113,122,130,139,149,158,168,178,189,200,210,222,234,246,

%U 258,270,284,297,310,324,338,353,367,382,398,413,429,445,462

%N The degree of polynomials related to Somos-6 sequences. Also for n > 4 the degree of the (n-5)-th involution in a family of involutions in the Cremona group of rank 5 defined by a Somos-6 sequence.

%C Let s(0), s(1), ..., s(4), s(5) be the 6 initial values in a Somos-6 sequence. The following terms s(6), s(7), ... are rational expressions in the 6 initial values derived from the Somos-6 recurrence: s(n) = ( s(n-1)*s(n-5) + s(n-2)*s(n-4) + s(n-3)^2 ) / s(n-6). E.g., s(6) = (s(1)*s(5) + s(2)*s(4) + s(3)^2) / s(0), s(7) = ... .

%C Because of the Laurent property of a Somos-6 sequence the denominator of these terms is a monomial in the initial values.

%C With the sequence e(n) = A369113(n), the tropical version of the Somos-6 sequence, the monomial D(n) is defined as Product_{k=0..5} s(k)^a(n-k). Define the polynomial G(n) to be s(n) * D(n). G(n) is 1 for n < 4, else G(n) is the numerator of s(n), so ..., G(4) = 1, G(5) = 1, G(6) = s(1)*s(5) + s(2)*s(4) + s(3)^2, ...

%C 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 This Somos-6 sequence defines a family (proposed name: Somos family) S of (birational) involutions in Cr_5(R), the Cremona group of rank 5.

%C A Somos involution S(n) in this family is defined as S(n) : RP^5 -> RP^5, (s(0) : s(1) : ... : s(4) : s(5)) -> (s(n+5) : s(n+4) : ... : s(n+1) : s(n)). For n > 0 S(n) = (G(n+5) : G(n+4)*m1 : ... : G(n+1)*m4 : G(n)*m5 ), with m1, m2, ..., m4, m5 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+5) and the term a(n+5) in the actual sequence.

%H S. Fomin and A. Zelevinsky, <a href="https://arxiv.org/abs/math/0104241">The Laurent phenomenon</a>, arXiv:math/0104241 [math.CO], 2001.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1,1,-2,2,-2,1).

%F a(n) = 1 + e(n-5) + e(n-4) + e(n-3) + e(n-2) + e(n-1) + e(n), where e(n) = A369113(n) is the exponent of one of the initial values in the denominator of s(n).

%F The growth rate is quadratic, a(n) = (3/20) * n^2 + O(n).

%F For n > 5 a(n) = (A212979(n-3) - 4) / 3. - _Helmut Ruhland_, Jan 31 2024

%o (Maxima) (Maxima) N : 6$ Len : 12$ /* Somos-N, N >= 2, Len = length of the calculated lists */

%o NofRT : floor (N / 2)$ /* number of terms in a Somos-N recurrence */

%o S : makelist (0, Len)$

%o G : makelist (0, Len)$ DegG : makelist (0, Len)$ /* G, the numerator of s() */

%o for i: 1 thru N do ( S[i] : s[i - 1], G[i] : 1, DegG[i] : 0 )$

%o for i: N + 1 thru Len do (

%o SS : 0,

%o for j : 1 thru NofRT do (

%o SS : SS + S[i - j] * S[i - N + j]

%o ),

%o S[i] : factor (SS / S[i - N]), G[i] : num (S[i]),

%o /* for N > 3 G is a homogenous polynomial, take the first monomial to determine the degree */

%o Mon : G[i], if N > 3 then ( Mon : args (Mon)[1] ),

%o DegG[i] : 0, for j : 0 thru N - 1 do ( DegG[i] : DegG[i] + hipow (Mon, s[j]) )

%o )$ DegG;

%Y Cf. A368052, A368481, A368483, A006722, A369113, A212979.

%K nonn

%O 0,7

%A _Helmut Ruhland_, Dec 26 2023