Helmut Ruhland - cp's OEIS frontend

User: Helmut Ruhland

Helmut Ruhland has authored 6 sequences.

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.

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).

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

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

A369113 Tropical version of Somos-6 sequence A006722.

Original entry on oeis.org

-1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 24, 25, 27, 29, 30, 32, 34, 36, 38, 39, 42, 44, 46, 48, 50, 53, 55, 57, 60, 62, 65, 67, 70, 73, 75, 78, 81, 84, 87, 89, 93, 96, 99, 102, 105, 109, 112, 115, 119, 122, 126, 129, 133, 137, 140, 144, 148, 152

Offset: 0

Helmut Ruhland, Jan 13 2024

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

Second difference has period 20.

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

Cf. A006722.

N : 6$ 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-5), a(n-2) + a(n-4), 2*a(n-3) ) - a(n-6) for all n in Z.

G.f.: (-1 + x + x^4)/((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x + x^2 + x^3 + x^4)). - Stefano Spezia, Jan 14 2024

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.

Original entry on oeis.org

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

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.

Original entry on oeis.org

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

cp's OEIS Frontend

User: Helmut Ruhland

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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A369611 Tropical version of Somos-7 sequence A006723.

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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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A369113 Tropical version of Somos-6 sequence A006722.

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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.

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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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