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A368052
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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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4
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0, 0, 0, 0, 2, 3, 5, 8, 10, 14, 18, 22, 28, 33, 39, 46, 52, 60, 68, 76, 86, 95, 105, 116, 126, 138, 150, 162, 176, 189, 203, 218, 232, 248, 264, 280, 298, 315, 333, 352, 370, 390, 410, 430, 452, 473, 495, 518, 540, 564, 588, 612, 638, 663, 689, 716, 742, 770
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OFFSET
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0,5
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COMMENTS
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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) = ... .
Because of the Laurent property of a Somos-4 sequence the denominator of these terms is a monomial in the initial values.
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, ...
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-4 sequence defines a family (proposed Somos family) S of (birational) involutions in Cr_3(R), the Cremona group of rank 3.
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.
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LINKS
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FORMULA
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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
G.f.: x^4*(2 - 3*x + 2*x^2)/((1 - x)^3*(1 + x^4)). - Michael Somos, Dec 27 2023
The growth rate is quadratic, a(n) = (1/4) * n^2 + O(n).
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
n mod 8 = 0 1 2 3 4 5 6 7
f8(n) = -8 -1 0 -8 10 3 2 10 (End)
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MATHEMATICA
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s[n_?(#>3 &)] := s[n] = Together[(s[n-1] s[n-3] + s[n-2]^2) / s[n-4]];
a[0|1|2|3] = 0; a[n_] := Exponent[Numerator[s[n]] /. {s[_] :> s}, s];
Table[a[n], {n, 0, 10}] (* or *)
LinearRecurrence[{3, -3, 1, -1, 3, -3, 1}, {0, 0, 0, 0, 2, 3, 5}, 40] (* Andrey Zabolotskiy, Dec 22 2023 *)
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 *)
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PROG
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(Maxima) N : 4$ Len : 12$ /* 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;
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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