A369611 Tropical version of Somos-7 sequence A006723.

-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

Author

Helmut Ruhland, Jan 27 2024

Keywords

• sign
• easy

Comments

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.

Links

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

Crossrefs

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

Programs

Maxima

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;

Formula

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