A333251 -id:A333251 - cp's OEIS frontend

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.

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

Helmut Ruhland, Dec 26 2023

nonn

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.

S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,0,1,-1,-1,1).

Cf. A368052, A368482, A368483, A006721, A333251.

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);

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

A369611 Tropical version of Somos-7 sequence A006723.

Original entry on oeis.org

-1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 31, 32, 33, 35, 37, 38, 40, 41, 43, 45, 47, 48, 50, 52, 54, 56, 58, 59, 62, 64, 66, 68, 70, 72, 75, 77, 79, 81, 84, 86, 89

Offset: 0

Helmut Ruhland, Jan 27 2024

Given the Somos-7 sequence with variables s(1), s(2), s(3), s(4), s(5), s(6), s(7) and recursion s(n) = (s(n-1)*s(n-6) + s(n-2)*s(n-5) + s(n-3)*s(n-4))/s(n-7), then s(n) is a Laurent polynomial in the variables with the numerator being irreducible and the denominator is Product_{k=0..6} s(k+1)^a(n-k).

Second difference has period 30.

Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,0,-1,-1,0,1).

Cf. A006723, A220838, A333251, A369113, A368483.

N : 7$ Len : 50$  /* tropical version of Somos-N, 2 <= N <= 7, Len = length of the calculated list */
NofRT : floor (N / 2)$  /* number of terms in a Somos-N recurrence */
A : makelist (0, Len)$  A[1] : -1$ for i: 2 thru N do ( A[i] : 0 )$
for i: N + 1 thru Len do (
   M : minf, for j : 1 thru NofRT do ( M : max ( M, A[i - j] + A[i - N + j] ) ),
   A[i] : M - A[i - N]
)$ A;

a(n) = max( a(n-1) + a(n-6), a(n-2) + a(n-5), a(n-4) + a(n-3) ) - a(n-7) for all n in Z.

G.f.: ( 1-x^2-x^3 ) / ( (1+x)*(1+x+x^2)*(x^4+x^3+x^2+x+1)*(x-1)^3 ). - R. J. Mathar, Jan 28 2024

cp's OEIS Frontend

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.

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