A368052 -id:A368052 - cp's OEIS frontend

A368482 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.

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

Offset: 0

Helmut Ruhland, Dec 26 2023

nonn

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.

S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1,1,-2,2,-2,1).

Cf. A368052, A368481, A368483, A006722, A369113, A212979.

(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;

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).
For n > 5 a(n) = (A212979(n-3) - 4) / 3. - Helmut Ruhland, Jan 31 2024

A368483 The degree of polynomials related to Somos-7 sequences. Also for n > 6 the degree of the (n-6)-th involution in a family of involutions in the Cremona group of rank 6 defined by a Somos-7 sequence.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 7, 9, 12, 14, 16, 19, 23, 26, 30, 33, 37, 42, 47, 51, 56, 61, 67, 73, 79, 84, 91, 98, 105, 112, 119, 126, 135, 143, 151, 159, 168, 177, 187, 196, 205, 215, 226, 236, 247, 257, 268, 280, 292, 303, 315, 327, 340, 353, 366

Offset: 0

Helmut Ruhland, Dec 26 2023

Let s(0), s(1), ..., s(5), s(6) be the 7 initial values in a Somos-7 sequence. The following terms s(7), s(8), ... are rational expressions in the 7 initial values derived from the Somos-7 recurrence: s(n) = ( s(n-1)*s(n-6) + s(n-2)*s(n-5) + s(n-3)*s(n-4) ) / s(n-7). E.g., s(7) = (s(1)*s(6) + s(2)*s(5) + s(3)*s(4)) / s(0), s(8) = ... .
Because of the Laurent property of a Somos-7 sequence the denominator of these terms is a monomial in the initial values.
With the sequence e(n) = A369611(n), the tropical version of the Somos-7 sequence, the monomial D(n) is defined as Product_{k=0..6} s(k)^a(n-k). Define the polynomial G(n) to be s(n) * D(n). G(n) is 1 for n < 7, else G(n) is the numerator of s(n), so ..., G(5) = 1, G(6) = 1, G(7) = s(1)*s(6) + s(2)*s(5) + s(3)*s(4), ...
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-7 sequence defines a family (proposed name: Somos family) S of (birational) involutions in Cr_6(R), the Cremona group of rank 6.
A Somos involution S(n) in this family is defined as S(n) : RP^6 -> RP^6, (s(0) : s(1) : ... : s(5) : s(6)) -> (s(n+6) : s(n+5) : ... : s(n+1) : s(n)). For n > 0 S(n) = (G(n+6) : G(n+5)*m1 : ... : G(n+1)*m5 : G(n)*m6 ), with m1, m2, ..., m5, m6 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+6) and the term a(n+6) 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 (0,1,1,0,0,0,-1,-1,0,1).

Cf. A368052, A368481, A368482, A006723, A369611.

N : 7$ Len : 11$     /* 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-6) + e(n-5) + e(n-4) + e(n-3) + e(n-2) + e(n-1) + e(n), where e(n) = A369611(n), the tropical version of Somos-7, is the exponent of one of the initial values in the denominator of s(n).

The growth rate is quadratic, a(n) = (7/60) * n^2 + O(n).

A345118 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a basketweave pattern where all the multiple strands are of unit width, the horizontal ones appearing as 1 X 3 rectangles, while the vertical ones as unit area squares.

Original entry on oeis.org

0, 4, 11, 20, 34, 50, 69, 92, 116, 144, 175, 208, 246, 286, 329, 376, 424, 476, 531, 588, 650, 714, 781, 852, 924, 1000, 1079, 1160, 1246, 1334, 1425, 1520, 1616, 1716, 1819, 1924, 2034, 2146, 2261, 2380, 2500, 2624, 2751, 2880, 3014, 3150, 3289, 3432, 3576, 3724

Offset: 0

Stefano Spezia, Jun 08 2021

			Illustrations for n = 1..8:
        _           _ _          _ _ _
       |_|         |_|_|        |_ _ _|
                   |_ _|        |_|_|_|
                                |_ _ _|
    a(1) = 4     a(2) = 11     a(3) = 20
     _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
    |_ _|_|_|   |_ _|_|_ _|  |_|_|_ _ _|_|
    |_|_ _ _|   |_|_ _ _|_|  |_ _ _|_|_ _|
    |_ _|_|_|   |_ _|_|_ _|  |_|_|_ _ _|_|
    |_|_ _ _|   |_|_ _ _|_|  |_ _ _|_|_ _|
                |_ _|_|_ _|  |_|_|_ _ _|_|
                             |_ _ _|_|_ _|
    a(4) = 34    a(5) = 50     a(6) = 69
      _ _ _ _ _ _ _      _ _ _ _ _ _ _ _
     |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
     |_ _ _|_|_ _ _|    |_|_ _ _|_|_ _ _|
     |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
     |_ _ _|_|_ _ _|    |_|_ _ _|_|_ _ _|
     |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
     |_ _ _|_|_ _ _|    |_|_ _ _|_|_ _ _|
     |_|_|_ _ _|_|_|    |_ _|_|_ _ _|_|_|
                        |_|_ _ _|_|_ _ _|
        a(7) = 92           a(8) = 116

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,-1,3,-3,1).

Cf. A000035, A004773, A056594, A115067, A211014, A316316, A316317, A368052.

LinearRecurrence[{3,-3,1,-1,3,-3,1},{0,4,11,20,34,50,69},50]
a[ n_] := (3*n^2 + 5*n)/2 - (-1)^Floor[n/4]*Boole[Mod[n, 4] == 3]; (* Michael Somos, Jan 25 2024 *)

concat(0, Vec(x*(4 - x - x^2 + 3*x^3 + x^4)/((1 - x)^3*(1 + x^4)) + O(x^40))) \\ Felix Fröhlich, Jun 09 2021

{a(n) = (3*n^2 + 5*n)/2 - (-1)^(n\4)*(n%4==3)}; /* Michael Somos, Jan 25 2024 */

O.g.f.: x*(4 - x - x^2 + 3*x^3 + x^4)/((1 - x)^3*(1 + x^4)).
E.g.f.: (exp(x)*x*(8 + 3*x) + (-1)^(1/4)*(sinh((-1)^(1/4)*x) - sin((-1)^(1/4)*x)))/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-4) + 3*a(n-5) - 3*a(n-6) + a(n-7) for n > 6.
a(n) = (n*(5 + 3*n) - (1 - (-1)^n)*sin((n-1)*Pi/4))/2.
a(n) = A211014(n/2) - A000035(n)*A056594((n-3)/2).
a(2*n) = A211014(n).
a(k) = A115067(k+1) for k not congruent to 3 mod 4 (A004773).
From Helmut Ruhland, Jan 29 2024: (Start)
For n > 1: a(n) - (2 * A368052(n+2) + A368052(n+3)) * 2 is periodic for n mod 8, i.e. a(n) = (2 * A368052(n+2) + A368052(n+3)) * 2 + f8(n) with
n mod 8 =   0   1   2   3   4   5   6   7
f8(n)   =   0   0  -3  -2  -2  -2   1   0   (End)

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A368482 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maxima

Formula

A368483 The degree of polynomials related to Somos-7 sequences. Also for n > 6 the degree of the (n-6)-th involution in a family of involutions in the Cremona group of rank 6 defined by a Somos-7 sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maxima

Formula

A345118 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a basketweave pattern where all the multiple strands are of unit width, the horizontal ones appearing as 1 X 3 rectangles, while the vertical ones as unit area squares.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maxima

Formula