A006721 -id:A006721 - cp's OEIS frontend

A271955 Somos's sequence {b(8,n)} defined in comment in A078495: a(0)=a(1)=...=a(18)=1; for n>=19, a(n)=(a(n-1)a(n-18)+a(n-9)a(n-10))/a(n-19).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 30, 50, 80, 122, 178, 250, 340, 800, 1308, 1924, 2780, 4136, 6452, 10476, 17348, 28720, 61664, 179696, 439304, 910464, 1686704, 2905792, 4793624, 7753616, 12537856

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Apr 17 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..533

Cf. A006721, A078495, A268199, A271948, A271949, A271950, A271952, A271954.

[n le 19 select 1 else (Self(n-1)*Self(n-18)+Self(n-9)*Self(n-10))/Self(n-19): n in [1..60]]; // G. C. Greubel, Jul 30 2018

a[k_,n_]:=a[k,n]=If[n>2k+2,(a[k,(n-1)]*a[k,(n-2k-2)]+a[k,(n-k-1)]*a[k,(n-k-2)])/a[k,(n-2k-3)],1];
Map[a[8,#]&,Range[0,50]] (* Peter J. C. Moses, Apr 17 2016 *)

{a(n) = if(n<= 19, 1, (a(n-1)*a(n-18) + a(n-9)*a(n-10))/a(n-19))}; for(n=1,50, print1(a(n), ", ")) \\ G. C. Greubel, Jul 30 2018

A276130 a(0) = a(1) = a(2) = a(3) = a(4) = 1; for n>4, a(n) = ( a(n-1) +a(n-3) )*( a(n-2)+a(n-4) ) / a(n-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 10, 55, 649, 38881, 6414706, 24978826228, 2913605221297249, 112139525368095766797655, 8403341152380185679389503620974065, 146904111947501701959735285821948223340424963459227

Offset: 0

Bruno Langlois, Aug 21 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..22

Cf. A006721.

a(n) = if (n<=4, 1, (9-3*(-1)^n)/2*a(n-1)*a(n-3)-a(n-2)-a(n-4)); \\ Michel Marcus, Aug 27 2016

a(n) = (9-3*(-1)^n)/2*a(n-1)*a(n-3)-a(n-2)-a(n-4).

A276534 a(n) = a(n-1) * a(n-4) * (a(n-2) * a(n-3) + 1) / a(n-5), with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 12, 108, 10584, 27454896, 94148851006224, 246222177535609206635748240, 62371770277951054762478578990896212287188931341600, 3750595553941161278345366267513070968239986992860645038477600300348697171928615364721752014400

Offset: 0

Seiichi Manyama, Nov 16 2016

nonn

Inspired by Somos-5 sequence.
a(n) is an integer for n >= 0.
a(n+1)/a(n) is an integer for n >= 0.

			a(5) = a(4) * b(4) =  1 * 2 =   2,
a(6) = a(5) * b(5) =  2 * 2 =   4,
a(7) = a(6) * b(6) =  4 * 3 =  12,
a(8) = a(7) * b(7) = 12 * 9 = 108.

Seiichi Manyama, Table of n, a(n) for n = 0..17

Cf. A006721, A276535.

def A(k, n)
  a = Array.new(2 * k + 1, 1)
  ary = [1]
  while ary.size < n + 1
    i = 0
    k.downto(1){|j|
      i += 1
      i *= a[j] * a[-j]
    }
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end
def A276534(n)
  A(2, n)
end

a(n) * a(n-5) = a(n-1) * a(n-4) + a(n-1) * a(n-2) * a(n-3) * a(n-4).
a(4-n) = a(n).
Let b(n) = b(n-4) * (b(n-2) * (b(0) * b(1) * ... * b(n-3))^2 + 1) with b(0) = b(1) = b(2) = b(3) = 1, then a(n) = a(n-1) * b(n-1) = b(0) * b(1) * ... * b(n-1) for n > 0.

A333260 Number of terms in polynomial sequence s(n) = (xs(n-1)s(n-4) + ys(n-2)s(n-3))/s(n-5), with s(k) = 1 for k = 0..4.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 7, 11, 16, 23, 33, 46, 64, 84, 109, 143, 184, 228, 283, 351, 429, 515, 615, 734, 871, 1017, 1181, 1376, 1593, 1821, 2077, 2372, 2694, 3035, 3409, 3832, 4294, 4777, 5299, 5888, 6522, 7180, 7891, 8681, 9523, 10400, 11337, 12367, 13465

Offset: 0

Michael Somos, Mar 13 2020

nonn

s(n) is a generalized Somos-5 sequence (A006721) having coefficients x, y in the recurrence numerator sum of products.

			a(7) = 4 because s(7) = x^3 + x^2*y + 2*x*y*z + y^2*z has 4 terms.

Cf. A006721.

a[ n_] := If[0 <= n <= 4, 1, RecurrenceTable[{s[m]*s[m - 5] == x*s[m - 1]*s[m - 4] + y*s[m - 2]*s[m - 3], s[0] == s[1] == s[2] == s[3] == s[4] == 1}, s, {m, Max[n, 4 - n]}] // Last // Factor // Expand // Length];

a(n) = a(4-n) for all n in Z.

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

A058058 Generalized Somos-7 sequence: a(n)a(n+7) = 3a(n+1)a(n+6) - 4a(n+2)* a(n+5) + 4a(n+3)a(n+4).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 9, 19, 33, 131, 681, 3921, 23801, 132881, 598121, 7466321, 141401273, 1865484899, 19358862929, 314151573363, 7831607063961, 237725833277411, 8937694547422641, 293153245305595201, 7098035759907924561, 310194702846756799041, 35075042744420641281841

Offset: 1

Robert G. Wilson v, Nov 20 2000

nonn

N. Elkies, posting to the NMBRTHRY(AT)LISTSERV.NODAK.EDU newsgroup, Nov. 2000.

G. C. Greubel, Table of n, a(n) for n = 1..170

Cf. A006720, A006721, A006722 and A006723.

I:=[1,1,1,1,1,1,1]; [n le 7 select I[n] else (3*Self(n-1)*Self(n-6) - 4*Self(n-2)*Self(n-5) + 4*Self(n-3)*Self(n-4))/Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 21 2018

a[1] =a[2] =a[3] =a[4] =a[5] =a[6] =a[7] =1; a[n_]:= a[n] = (3*a[n-1]*a[n-6] - 4*a[n-2]*a[n-5] + 4*a[n-3]*a[n-4])/a[n-7]; Table[ a[n], {n, 1, 35}]

{a(n) = if (n <=7, 1, (3*a(n-1)*a(n-6) - 4*a(n-2)*a(n-5) + 4*a(n-3)*a(n-4))/a(n-7))};
for(n=1, 30, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018

A078530 Bilinear recursive sequence.

Original entry on oeis.org

0, 3, 1, 1, 1, 1, 2, 3, 9, 27, 81, 729, 0, 59049, -531441, 14348907, -387420489, 10460353203, -564859072962, 22876792454961, -1853020188851841, 150094635296999121, -12157665459056928801, 2954312706550833698643, 0

Offset: 0

Michael Somos, Nov 25 2002

Cf. A078529, A006720, A006721.

a[ n_] := With[{m = Mod[n, 12]}, Sign[m] * 2^Boole[m==6] * (-1)^(Mod[Floor[n/12], 2]*(n-1)) * 3^(Boole[m==0] + Floor[(n-4)^2/8])]; (* Michael Somos, Dec 10 2023 *)

{a(n) = sign(n%12) * (1 + (n%12==6)) * (-1)^(n\12%2 * (n-1)) * 3^((n%12==0) + (n-4)^2\8)};

a(n) * a(n-8) = 81 * (a(n-2)*a(n-6) - 2*a(n-4)^2).
0 = a(n) * a(n-5) + 3 * a(n-1) * a(n-4) - 9 * a(n-2)*a(n-3).
a(12*n) = 0.
a(2*n+1) = a(-2*n+7) = a(4*n+2)/(81^(n-1)*(a(2*n-1)*a(2*n+2)^2 - a(2*n+3)*a(2*n)^2)) for all n in Z. - Michael Somos, Dec 10 2023
a(n+12) = -(-27)^(n+2) * a(n) for all n in Z. - Michael Somos, Dec 11 2023

A105236 a(n+5) = (a(n+4)a(n+1) + 2a(n+3)*a(n+2))/a(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 5, 11, 41, 233, 689, 5337, 49081, 458299, 3603685, 93208147, 1476087601, 27470407569, 816413467841, 43620306030449, 1172020019840081, 70063780891581107, 5804382690927311525, 511286588817798535899

Offset: 0

Andrew Hone, Apr 14 2005

nonn

This is a bilinear recurrence of Somos 5 type, hence the terms a(n) are associated with a sequence of points P_n = P_0 + n*P on an elliptic curve E. In this case the curve E has integral j-invariant j=10976.

Seiichi Manyama, Table of n, a(n) for n = 0..147
A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences, arXiv:math/0501554 [math.NT], 2005-2006.
A. N. W. Hone, Bilinear recurrences and addition formulas for hyperelliptic sigma functions, arXiv:math/0501162 [math.NT], 2005.
A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.
A. J. van der Poorten, Elliptic curves and continued fractions, J. Int. Sequences, Volume 8, no. 2 (2005), article 05.2.5.

Cf. A006720, A006721.

[n le 5 select 1 else (Self(n-1)*Self(n-4) +2*Self(n-2)*Self(n-3))/Self(n-5): n in [1..41]]; // G. C. Greubel, Nov 26 2022

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==1,a[n]==(2 a[-3+n] a[-2+n]+a[-4+n] a[-1+n])/a[-5+n]},a,{n,30}] (* Harvey P. Dale, Sep 15 2013 *)

@CachedFunction
def a(n): # a = A105236
  if (n<5): return 1
  else: return (a(n-1)*a(n-4) +2*a(n-2)*a(n-3))/a(n-5)
[a(n) for n in range(41)] # G. C. Greubel, Nov 26 2022

A129982 Fibonacci numbers sandwiched between 1's.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 1, 5, 1, 8, 1, 13, 1, 21, 1, 34, 1, 55, 1, 89, 1, 144, 1, 233, 1, 377, 1, 610, 1, 987, 1, 1597, 1, 2584, 1, 4181, 1, 6765, 1, 10946, 1, 17711, 1, 28657, 1, 46368, 1, 75025, 1, 121393, 1, 196418, 1, 317811, 1, 514229, 1, 832040, 1, 1346269, 1

Offset: 0

Paul Curtz, Jun 14 2007

This sequence is similar to Somos-5 (A006721). - Michael Somos, Aug 15 2014

			G.f. = 1 + x^2 + x^3 + x^4 + x^5 + x^6 + 2*x^7 + x^8 + 3*x^9 + x^10 + ...

Index entries for linear recurrences with constant coefficients, signature (0,2,0,0,0,-1).

Cf. A000045, A006721.

G := 1/(1-x^2)+x^3/(1-x^2-x^4); Gser := series(G, x = 0, 70); seq(coeff(Gser, x, n), n = 0 .. 65); # Emeric Deutsch, Jul 09 2007

a[ n_] := If[ OddQ[n], Fibonacci[ Quotient[ n, 2]], 1]; (* Michael Somos, Aug 15 2014 *)

{a(n) = if( n%2, fibonacci( n\2), 1)}; /* Michael Somos, Aug 15 2014 */

G.f.: (1 - x^2 + x^3 - x^4 - x^5) / (1 - 2*x^2 + x^6). - Michael Somos, Aug 15 2014
a(2-n) = (-1)^(mod(n, 4) == 1) * a(n) for all n in Z. - Michael Somos, Aug 15 2014
a(2*n) = 1, a(2*n + 1) = A000045(n) for all n in Z. - Michael Somos, Aug 15 2014
a(n) = 2*a(n-2) - a(n-6) for all n in Z. - Michael Somos, Aug 15 2014
0 = a(n)*a(n+5) - a(n+1)*a(n+4) - a(n+2)*a(n+3) for all even n in Z. - Michael Somos, Aug 15 2014
0 = a(n)*a(n+5) - a(n+1)*a(n+4) + a(n+2)*a(n+3) for all odd n in Z. - Michael Somos, Aug 15 2014

More terms from Emeric Deutsch, Jul 09 2007

A272038 Somos's sequence {b(9,n)} defined in comment in A078495: a(0)=a(1)=...=a(20)=1; for n>=21, a(n)=(a(n-1)a(n-20)+a(n-10)a(n-11))/a(n-21).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 19, 31, 51, 81, 123, 179, 251, 341, 451, 1045, 1691, 2451, 3459, 4977, 7467, 11679, 18755, 30349, 48763, 100474, 282777, 679512, 1391391, 2547414, 4327101

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Apr 18 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..578

Cf. A006721, A078495, A268199, A271954, A271948, A271949, A271950, A271952, A271955.

def b(k, n)
  b = Array.new(2 * k + 3, 1)
  (2 * k + 3..n).each{|i|
    j = (b[i - 1] * b[i - 2 * k - 2] + b[i - k - 1] * b[i - k - 2]) / b[i - 2 * k - 3].to_r
    j = j.to_i if j.denominator == 1
    b[i] = j
  }
  b[0..n]
end
p b(9, n) # Seiichi Manyama, May 04 2016

cp's OEIS Frontend

A271955 Somos's sequence {b(8,n)} defined in comment in A078495: a(0)=a(1)=...=a(18)=1; for n>=19, a(n)=(a(n-1)*a(n-18)+a(n-9)*a(n-10))/a(n-19).

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A276130 a(0) = a(1) = a(2) = a(3) = a(4) = 1; for n>4, a(n) = ( a(n-1) +a(n-3) )*( a(n-2)+a(n-4) ) / a(n-5).

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A276534 a(n) = a(n-1) * a(n-4) * (a(n-2) * a(n-3) + 1) / a(n-5), with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

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A333260 Number of terms in polynomial sequence s(n) = (x*s(n-1)*s(n-4) + y*s(n-2)*s(n-3))/s(n-5), with s(k) = 1 for k = 0..4.

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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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A058058 Generalized Somos-7 sequence: a(n)*a(n+7) = 3*a(n+1)*a(n+6) - 4*a(n+2)* a(n+5) + 4*a(n+3)*a(n+4).

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A078530 Bilinear recursive sequence.

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A105236 a(n+5) = (a(n+4)*a(n+1) + 2*a(n+3)*a(n+2))/a(n).

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A271955 Somos's sequence {b(8,n)} defined in comment in A078495: a(0)=a(1)=...=a(18)=1; for n>=19, a(n)=(a(n-1)a(n-18)+a(n-9)a(n-10))/a(n-19).

A333260 Number of terms in polynomial sequence s(n) = (xs(n-1)s(n-4) + ys(n-2)s(n-3))/s(n-5), with s(k) = 1 for k = 0..4.

A058058 Generalized Somos-7 sequence: a(n)a(n+7) = 3a(n+1)a(n+6) - 4a(n+2)* a(n+5) + 4a(n+3)a(n+4).

A105236 a(n+5) = (a(n+4)a(n+1) + 2a(n+3)*a(n+2))/a(n).

A272038 Somos's sequence {b(9,n)} defined in comment in A078495: a(0)=a(1)=...=a(20)=1; for n>=21, a(n)=(a(n-1)a(n-20)+a(n-10)a(n-11))/a(n-21).