A220838 -id:A220838 - cp's OEIS frontend

A368052 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.

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

Offset: 0

Helmut Ruhland, Dec 09 2023

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.

S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,-1,3,-3,1).

Cf. A006720, A220838, A345118.

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 *)

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;

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
From Helmut Ruhland, Jan 29 2024: (Start)
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)

More terms from Andrey Zabolotskiy, Dec 22 2023

A236294 a(n) = max( a(n-1) + a(n-3), 2*a(n-2) ) - a(n-4), with a(0)=1, a(1)=1, a(2)=2, a(3)=3.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 22, 25, 27, 30, 33, 35, 39, 42, 45, 49, 52, 56, 60, 63, 68, 72, 76, 81, 85, 90, 95, 99, 105, 110, 115, 121, 126, 132, 138, 143, 150, 156, 162, 169, 175, 182, 189, 195, 203, 210, 217, 225, 232, 240, 248, 255

Offset: 0

Michael Somos, Jan 21 2014

Tropical version of Somos-4 sequence A006720.
Second difference is period 8 sequence [1, 0, -1, 2, -1, 0, 1, -1, ...]
The numerator of the g.f. is the reciprocal polynomial of the numerator of the g.f. of A220838.

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 9*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. A006720, A220838.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^2-x^4+2*x^5-x^6)/((1-x)^2*(1-x^8)))); // G. C. Greubel, Aug 07 2018

CoefficientList[Series[(1-x+x^2-x^4+2*x^5-x^6)/((1-x)^2*(1-x^8)), {x, 0, 50}], x] (* G. C. Greubel, Aug 07 2018 *)

{a(n) = if( n<-4, n = -8-n); if( n<0, -(n==-4), polcoeff( (1 - x + x^2 - x^4 + 2*x^5 - x^6) / ( (1 - x)^2 * (1 - x^8) ) + x * O(x^n), n))};

G.f.: (1 - x + x^2 - x^4 + 2*x^5 - x^6) / ( (1 - x)^2 * (1 - x^8) ).
a(n) = a(-8 - n) = A220838(n + 5) for all n in Z.
0 = (a(n+5) - 2*a(n+3) + a(n+1)) * (a(n+4) - 2*a(n+2) * a(n)) for all n in Z.

A333251 Tropical version of Somos-5 sequence A006721.

Original entry on oeis.org

-1, 0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 22, 24, 26, 27, 30, 32, 34, 36, 39, 41, 43, 46, 49, 51, 54, 57, 60, 62, 66, 69, 72, 75, 79, 82, 85, 89, 93, 96, 100, 104, 108, 111, 116, 120, 124, 128, 133, 137, 141, 146, 151

Offset: 0

Michael Somos, Mar 13 2020

If (x, y, s(0), .., s(4)) are 7 variables and s(n) = (x*s(n-1)*s(n-4) + y*s(n-2)*s(n-3))/s(n-5) for n>=5 is the generalized Somos-5 sequence, then s(n) is a Laurent polynomial in the variables with the numerator being irreducible and the denominator is Product_{k=0..4} s(k)^a(n-k).

			G.f. = -1 + x^5 + x^6 + x^7 + 2*x^8 + 3*x^9 + 3*x^10 + 4*x^11 + ...
s(7) = ((s(0)*s(3)^2*s(4) + s(1)^2*s(4)^2)*x*y + s(1)*s(2)*s(3)*s(4)*(y^2+x^3) + s(2)^2*s(3)^2*x^2*y)/(s(0)^1*s(1)^1*s(2)^1*s(3)^0*s(4)^0).

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

Cf. A006721, A220838.

a[ n_] := With[{m = Mod[n, 14]}, Quotient[n^2, 28] - Boole[m==0] + Boole[m==5] + Boole[m==9]];
a[ n_] := SeriesCoefficient[ -(1 - x - x^2 + x^3 - x^5) / ((1 - x) * (1 - x^2) * (1 - x^7)), {x, 0, Abs@n}];

{a(n) = n^2\28 - (n%14==0) + (n%14==5) + (n%14==9)};

{a(n) = n=abs(n); polcoeff( -(1 - x - x^2 + x^3 - x^5) / ((1 - x) * (1 - x^2) * (1 - x^7)) + x * O(x^n), n)};

G.f.: -(1 - x - x^2 + x^3 - x^5)/((1 - x)*(1 - x^2)*(1 - x^7)).
a(n) = max( a(n-1) + a(n-4), a(n-2) + a(n-3) ) - a(n-5) for all n in Z.
a(n) = a(n+7) - 2 - floor(n/2) for all n in Z.
Second difference has period 14.

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

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A368052 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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A236294 a(n) = max( a(n-1) + a(n-3), 2*a(n-2) ) - a(n-4), with a(0)=1, a(1)=1, a(2)=2, a(3)=3.

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A333251 Tropical version of Somos-5 sequence A006721.

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A369611 Tropical version of Somos-7 sequence A006723.

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