A368482 -id:A368482 - cp's OEIS frontend

A212979 Number of (w,x,y) with all terms in {0,...,n} and range=average.

1, 1, 1, 7, 10, 13, 19, 25, 34, 40, 49, 61, 70, 82, 94, 109, 124, 136, 154, 172, 190, 208, 226, 250, 271, 292, 316, 340, 367, 391, 418, 448, 475, 505, 535, 568, 601, 631, 667, 703, 739, 775, 811, 853, 892, 931, 973, 1015, 1060, 1102, 1147, 1195, 1240

Offset: 0

Clark Kimberling, Jun 03 2012

For a guide to related sequences, see A212959.

			a(3)=7 counts these (w,x,y): (0,0,0) and the six permutations of (1,2,3).
G.f. = 1 + x + x^2 + 7*x^3 + 10*x^4 + 13*x^5 + 19*x^6 + 25*x^7 + 34*x^8 + ... - _Michael Somos_, Jan 25 2024

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1,1,-2,2,-2,1).

Cf. A212959, A368482.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y] - Min[w, x, y] == (w + x + y)/3, s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212979 *)
a[ n_] := If[n<0, a[-1-n], Sum[ Boole[Max[t] - Min[t] == Mean[t]], {t, Tuples[Range[0, n], 3]}]]; (* Michael Somos, Jan 25 2024 *)
a[ n_] := (9*(n^2+n) + 6*{10, 7, 1, 12, 10, 5, 7, 6, 12, 5}[[1 + Min[Mod[n, 20], Mod[-1-n, 20]]]])/20 - 2; (* Michael Somos, Jan 25 2024 *)

{a(n) = (9*(n^2+n) + 6*[10, 7, 1, 12, 10, 5, 7, 6, 12, 5][1 + min(n%20, (-1-n)%20)])/20 - 2}; /* Michael Somos, Jan 25 2024 */

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) - 2*a(n-6) + 2*a(n-7) - 2*a(n-8) + a(n-9).
G.f.: (1 - x + x^2 + 5*x^3 - 3*x^4 + 5*x^5 + x^6 - x^7 + x^8)/(1 - 2*x + 2*x^2 - 2*x^3 + x^4 - x^5 + 2*x^6 - 2*x^7 + 2*x^8 - x^9).
From Michael Somos, Jan 25 2024: (Start)
a(n) = a(-1-n) for all n in Z.
G.f.: (1 + x)*(1 - x + x^2 + 5*x^3 - 3*x^4 + 5*x^5 + x^6 - x^7 + x^8)/((1 - x)*(1 - x^4)*(1 - x^5)). (End)
For n > 2, a(n) = 3 * A368482(n+3) + 4. - 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).

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

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A212979 Number of (w,x,y) with all terms in {0,...,n} and range=average.

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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.

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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.

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