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A368481
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.
2
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, 103, 112, 120, 130, 140, 150, 160, 172, 183, 194, 206, 219, 231, 244, 258, 272, 285, 300, 315, 330, 345, 362, 378, 394, 411, 429, 446, 464, 483, 502, 520, 540, 560, 580, 600, 622, 643, 664, 686, 709
OFFSET
0,6
COMMENTS
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) = ... .
Because of the Laurent property of a Somos-5 sequence the denominator of these terms is a monomial in the initial values.
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), ...
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-5 sequence defines a family (proposed Somos family) S of (birational) involutions in Cr_4(R), the Cremona group of rank 4.
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.
LINKS
S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
FORMULA
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
The growth rate is quadratic, a(n) = (5/28) * n^2 + O(n).
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
PROG
(Maxima) N : 5$ Len : 15$ /* 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])
)
)$
args (DegG);
CROSSREFS
KEYWORD
nonn
AUTHOR
Helmut Ruhland, Dec 26 2023
STATUS
approved