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%I #38 Feb 07 2024 09:03:01
%S 0,0,0,0,0,2,3,4,6,9,11,14,18,22,25,30,35,40,45,52,58,64,71,79,86,94,
%T 103,112,120,130,140,150,160,172,183,194,206,219,231,244,258,272,285,
%U 300,315,330,345,362,378,394,411,429,446,464,483,502,520,540,560,580,600,622,643,664,686,709
%N The degree of polynomials related to Somos-5 sequences. Also for n > 4 the degree of the (n-4)-th involution in a family of involutions in the Cremona group of rank 4 defined by a Somos-5 sequence.
%C Let s(0), s(1), s(2), s(3), s(4) be the 5 initial values in a Somos-5 sequence. The following terms s(5), s(6), ... are rational expressions in the 5 initial values derived from the Somos-5 recurrence: s(n) = ( s(n-1)*s(n-4) + s(n-2)*s(n-3) ) / s(n-5). E.g., s(5) = (s(1)*s(4) + s(2)*s(3)) / s(0), s(6) = ... .
%C Because of the Laurent property of a Somos-5 sequence the denominator of these terms is a monomial in the initial values.
%C With the sequence e(n) = A333251(n), the tropical version of the Somos-5 sequence, the monomial D(n) is defined as Product_{k=0..4} s(k)^a(n-k). Define the polynomial G(n) to be s(n) * D(n). G(n) is 1 for n < 5, else G(n) is the numerator of s(n), so ..., G(3) = 1, G(4) = 1, G(5) = s(1)*s(4) + s(2)*s(3), ...
%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-5 sequence defines a family (proposed Somos family) S of (birational) involutions in Cr_4(R), the Cremona group of rank 4.
%C A Somos involution S(n) in this family is defined as S(n) : RP^4 -> RP^4, (s(0) : s(1) : s(2) : s(3) : s(4)) -> (s(n+4) : s(n+3) : s(n+2) : s(n+1) : s(n)). For n > 0 S(n) = (G(n+4) : G(n+3)*m1 : G(n+2)*m2 : G(n+1)*m3 : G(n)*m4 ), with m1, m2, m3, m4 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+4) and the term a(n+4) 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_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,0,0,1,-1,-1,1).
%F a(n) = 1 + e(n-4) + e(n-3) + e(n-2) + e(n-1) + e(n), where e(n) = A333251(n) is the exponent of one of the initial values in the denominator of s(n). - _Andrey Zabolotskiy_ Jan 09 2024
%F The growth rate is quadratic, a(n) = (5/28) * n^2 + O(n).
%F G.f.: x^5 * (2+x-x^2+x^3+2*x^4) / ( (1-x)^3 * (x+1) * (x^6+x^5+x^4+x^3+x^2+x+1) ). - _Joerg Arndt_, Jan 14 2024
%o (Maxima) N : 5$ Len : 15$ /* 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 )
%o )$
%o args (DegG);
%Y Cf. A368052, A368482, A368483, A006721, A333251.
%K nonn
%O 0,6
%A _Helmut Ruhland_, Dec 26 2023