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A368482
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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.
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3
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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, 83, 90, 97, 105, 113, 122, 130, 139, 149, 158, 168, 178, 189, 200, 210, 222, 234, 246, 258, 270, 284, 297, 310, 324, 338, 353, 367, 382, 398, 413, 429, 445, 462
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OFFSET
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0,7
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COMMENTS
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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) = ... .
Because of the Laurent property of a Somos-6 sequence the denominator of these terms is a monomial in the initial values.
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, ...
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-6 sequence defines a family (proposed name: Somos family) S of (birational) involutions in Cr_5(R), the Cremona group of rank 5.
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.
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LINKS
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FORMULA
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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).
The growth rate is quadratic, a(n) = (3/20) * n^2 + O(n).
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PROG
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(Maxima) (Maxima) N : 6$ 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
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AUTHOR
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STATUS
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approved
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