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The degree of polynomials related to Somos-4 sequences. Also for n > 2 the degree of the (n-3)-th involution in a family of involutions in the Cremona group of rank 3 defined by a Somos-4 sequence.
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%I #82 Feb 02 2024 07:47:33

%S 0,0,0,0,2,3,5,8,10,14,18,22,28,33,39,46,52,60,68,76,86,95,105,116,

%T 126,138,150,162,176,189,203,218,232,248,264,280,298,315,333,352,370,

%U 390,410,430,452,473,495,518,540,564,588,612,638,663,689,716,742,770

%N The degree of polynomials related to Somos-4 sequences. Also for n > 2 the degree of the (n-3)-th involution in a family of involutions in the Cremona group of rank 3 defined by a Somos-4 sequence.

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

%C Because of the Laurent property of a Somos-4 sequence the denominator of these terms is a monomial in the initial values.

%C With the sequence e(n) = A220838(n), the tropical version of the Somos-4 sequence, the monomial D(n) is defined as Product_{k=0..3} s(k)^a(n+1-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(2) = 1, G(3) = 1, G(4) = s(1)*s(3) + s(2)^2, G(5) = s(0)*s(3)^2 + s(1)*s(2)*s(3) + s(2)^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-4 sequence defines a family (proposed Somos family) S of (birational) involutions in Cr_3(R), the Cremona group of rank 3.

%C A Somos involution S(n) in this family is defined as S(n) : RP^3 -> RP^3, (s(0) : s(1) : s(2) : s(3)) -> (s(n+3) : s(n+2) : s(n+1) : s(n)). For n > 0, S(n) = (G(n+3) : G(n+2)*m1 : G(n+1)*m2 : G(n)*m3 ), with m1, m2, m3 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+3) and the term a(n+3) 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_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,-1,3,-3,1).

%F a(n) = 1 + e(n-2) + e(n-1) + e(n) + e(n+1), where e(n) = A220838(n) is the exponent of one of the initial values in the denominator of s(n). - _Andrey Zabolotskiy_, Dec 22 2023

%F G.f.: x^4*(2 - 3*x + 2*x^2)/((1 - x)^3*(1 + x^4)). - _Michael Somos_, Dec 27 2023

%F From _Helmut Ruhland_, Jan 29 2024: (Start)

%F The growth rate is quadratic, a(n) = (1/4) * n^2 + O(n).

%F For n > 2: a(n) - (A345118(n-3) + 2 * A345118(n-2)) / 18 is periodic for n mod 8, i.e. a(n) = (A345118(n-3) + 2 * A345118(n-2) + f8(n)) / 18 with

%F n mod 8 = 0 1 2 3 4 5 6 7

%F f8(n) = -8 -1 0 -8 10 3 2 10 (End)

%t s[n_?(#>3 &)] := s[n] = Together[(s[n-1] s[n-3] + s[n-2]^2) / s[n-4]];

%t a[0|1|2|3] = 0; a[n_] := Exponent[Numerator[s[n]] /. {s[_] :> s}, s];

%t Table[a[n], {n, 0, 10}] (* or *)

%t LinearRecurrence[{3, -3, 1, -1, 3, -3, 1}, {0, 0, 0, 0, 2, 3, 5}, 40] (* _Andrey Zabolotskiy_, Dec 22 2023 *)

%t a[ n_] := If[ n>=0, SeriesCoefficient[ x^4*(2 - 3*x + 2*x^2)/((1 - x)^3*(1 + x^4)), {x, 0, n}], a[3-n]]; (* _Michael Somos_, Dec 27 2023 *)

%o (Maxima) N : 4$ 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. A006720, A220838, A345118.

%K nonn,easy

%O 0,5

%A _Helmut Ruhland_, Dec 09 2023

%E More terms from _Andrey Zabolotskiy_, Dec 22 2023