A006721 -id:A006721 - cp's OEIS frontend

A242107 Reduced division polynomials associated with elliptic curve y^2 + x*y = x^3 - x^2 - x + 1 and multiples of point (0, 1).

0, 1, 1, 1, 1, -1, 2, -3, 1, -5, -7, -13, -16, 11, -57, 131, -113, 389, 670, 2311, 3983, 9, 23647, -81511, 140576, -484247, -833503, -5751815, -14871471, -17124617, -147165662, 710017141, -2273917871, 9644648819, 11396432249, 204006839259, 808162720720

Offset: 0

Michael Somos, Aug 15 2014

sign

This sequence is similar to Somos-5 (A006721).

For the elliptic curve "58a1" and point (0, 1) the multiple n*(0, 1) = ((3-(-1)^n)/2 * a(n+1)*a(n-1) / a(n)^2, a(n+2)^2 * a(n-4) / a(n)^3). - Michael Somos, Feb 23 2020

			a(9) = -5 and the point multiple 9*(0, 1) = (-14/(-5)^2, -169/(-5)^3).

Chai Wah Wu, Table of n, a(n) for n = 0..320
LMFDB, Elliptic Curve with LMFDB label 58.a1 (Cremona label 58a1).

Cf. A006721, A178622, A242108, A328380.

I:=[1,1,1,1,-1]; [0] cat [n le 5 select I[n] else (-Self(n-1)* Self(n-4) + Self(n-2)*Self(n-3))/Self(n-5): n in [1..30]]; // G. C. Greubel, Aug 05 2018

Join[{0}, RecurrenceTable[{a[n] == (-a[n-1]*a[n-4] + a[n-2]*a[n-3])/a[n-5], a[0] == 0, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == -1}, a, {n, 0, 50}]] (* G. C. Greubel, Aug 05 2018 *)

{a(n) = my(s=1, v); if( n<0, s=-1; n=-n); s^(n+1) * if( n, v = vector(n, k, 1); if( n<6, (-1)^(n>4), v[5] = -1; for(k=6, n, v[k] = (-v[k-1] * v[k-4] + v[k-2] * v[k-3]) / v[k-5]); v[n]))};

{a(n) = sign(n) * subst(elldivpol(ellinit([1, -1, 0, -1, 1]), abs(n)), x, 0) / (if(n%2, 1, 2) * (-1)^((n-1)\2) * 2^(n^2\4))}; /* Michael Somos, Feb 23 2020 */

{a(n) = my(E=ellinit([1, -1, 0, -1, 1]), z=ellpointtoz(E, [0, 1])); (-1)^(n\2) * round(ellsigma(E, n*z) / (ellsigma(E, z)^n^2 * 2^(n^2\4))) }; /* Michael Somos, Feb 25 2020 */

from gmpy2 import divexact
A242107 = [0,1,1,1,1,-1]
for n in range(6,30):
    A242107.append(int(divexact(-A242107[n-1]*A242107[n-4]+A242107[n-2]*A242107[n-3],A242107[n-5])))
print(A242107) # Chai Wah Wu, Aug 15 2014

a(n) = -(-1)^n * a(-n) for all n in Z.
0 = a(n)*a(n+5) + a(n+1)*a(n+4) - a(n+2)*a(n+3) for all n in Z.
0 = a(n)*a(n+7) - a(n+1)*a(n+6) - 2*a(n+2)*a(n+5) for all n in Z.
0 = a(n)*a(n+4) + a(n+1)*a(n+3) - a(n+2)*a(n+2) for all even n in Z.
0 = a(n)*a(n+4) + 2*a(n+1)*a(n+3) - a(n+2)*a(n+2) for all odd n in Z.
abs(a(n)) = A242108(n) for all n in Z.
a(2*n) = A178622(n) for all n in Z. - Michael Somos, Aug 21 2014
a(2*n-3) = A328380(n) for all n in Z. - Michael Somos, Feb 23 2020

Definition edited by Michael Somos, Feb 23 2020

A271950 Somos's sequence {b(5,n)} defined in comment in A078495: a(0)=a(1)=...=a(12)=1; for n>=13, a(n)=(a(n-1)a(n-12)+a(n-6)a(n-7))/a(n-13).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 15, 27, 47, 77, 119, 301, 519, 827, 1351, 2345, 4263, 10598, 35021, 91652, 200923, 396578, 742721, 2258305, 5126953, 14354017, 45716169, 138331649, 377080865, 1330892225, 5490413305, 16470110241

Offset: 0

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

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..392

Cf. A006721, A078495, A268199, A271948, A271949.

[n le 13 select 1 else ((Self(n-1)*Self(n-12) + Self(n-6)*Self(n- 7) )/Self(n-13)): n in [1..40]]; // G. C. Greubel, Feb 21 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[5,#]&,Range[0,50]] (* Peter J. C. Moses, Apr 17 2016 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==a[7]== a[8]== a[9]== a[10]== a[11] ==a[12]==1,a[n]==(a[n-1]a[n-12]+a[n-6]a[n-7])/a[n-13]},a,{n,50}] (* Harvey P. Dale, May 01 2018 *)

{a(n) = if(n< 12, 1, (a(n-1)*a(n-12) + a(n-6)*a(n-7))/a(n-13))};
for(n=0,40, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018

A271952 Somos's sequence {b(6,n)} defined in comment in A078495: a(0)=a(1)=...=a(14)=1; for n>=15, a(n)=(a(n-1)a(n-14)+a(n-7)a(n-8))/a(n-15).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 16, 28, 48, 78, 120, 176, 432, 728, 1120, 1736, 2832, 4864, 8576, 20224, 63808, 162624, 348224, 668288, 1204736, 2114560, 6175744, 13394432, 34860544, 104595968, 304683008, 807587840

Offset: 0

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

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..440

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

[n le 15 select 1 else (Self(n-1)*Self(n-14)+Self(n-7)*Self(n-8))/Self(n-15): 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[6,#]&,Range[0,50]] (* Peter J. C. Moses, Apr 17 2016 *)

{a(n) = if(n<= 15, 1, (a(n-1)*a(n-14) + a(n-7)*a(n-8))/a(n-15))}; for(n=1,50, print1(a(n), ", ")) \\ G. C. Greubel, Jul 30 2018

A210098 Somos-5 sequence variant: a(n) = (a(n-1) * a(n-4) - a(n-2) * a(n-3)) / a(n-5), a(0) = 0, a(1) = a(2) = a(3) = a(4) = 1, a(5) = 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, -1, -3, -5, -4, -11, -13, -7, 23, 86, 87, 199, 415, 799, -152, -4159, -8063, -17047, -38727, -155366, -142471, 445015, 2309453, 7627979, 13609844, 81138437, 187790979, 142104721, -1743980081, -12357952274, -25547499185, -134098256401

Offset: 0

Michael Somos, Mar 17 2012

sign

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).

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

Alois P. Heinz, Table of n, a(n) for n = 0..200
K. S. Brown, A Quasi-Periodic Sequence
Index to divisibility sequences

Cf. A006721.

I:=[1,1,1,1,2]; [n le 5 select I[n] else (Self(n-1)*Self(n-4) - Self(n-2)*Self(n-3))/Self(n-5): n in [1..50]]; // G. C. Greubel, Aug 11 2018

a:= proc(n) a(n):= `if`(n<6, [0, 1$4, 2][n+1],
      (a(n-1)*a(n-4) -a(n-2)*a(n-3)) / a(n-5))
    end:
seq (a(n), n=0..40);  # Alois P. Heinz, Oct 20 2012

Join[{0},RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[5]==2,a[n] == (a[n-1]a[n-4]-a[n-2]a[n-3])/a[n-5]},a,{n,40}]] (* Harvey P. Dale, Oct 20 2012 *)

{a(n) = my(v, m); if( n==0, 0, m = abs(n); sign(n) * if( m<6, 1 + (m>4), v = vector( m, i, 1); v[5] = 2; for( i=6, m, v[i] = (v[i-1] * v[i-4] - v[i-2] * v[i-3]) / v[i-5]); v[m]))};

a(n) = -a(-n), a(n) * a(n-5) = a(n-1) * a(n-4) - a(n-2) * a(n-3) for all n in Z.

a(n+4) * a(n-4) = a(n+2) * a(n-2) - a(n) * a(n), a(n+2) * a(n-2) = (2 - (-1)^n) * a(n+1) * a(n-1) - a(n) * a(n) for all n in Z.

A271954 Somos's sequence {b(7,n)} defined in comment in A078495: a(0)=a(1)=...=a(16)=1; for n>=17, a(n)=(a(n-1)a(n-16)+a(n-8)a(n-9))/a(n-17).

Original entry on oeis.org

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, 11, 17, 29, 49, 79, 121, 177, 249, 597, 989, 1483, 2209, 3425, 5589, 9447, 16137, 36240, 109683, 273382, 574885, 1081260, 1898415, 3213378, 5381793, 15251949, 31924773, 78189885

Offset: 0

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

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..487

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

[n le 17 select 1 else (Self(n-1)*Self(n-16)+Self(n-8)*Self(n-9))/Self(n-17): 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[7,#]&,Range[0,50]] (* Peter J. C. Moses, Apr 17 2016 *)

{a(n) = if(n<= 17, 1, (a(n-1)*a(n-16) + a(n-8)*a(n-9))/a(n-17))}; for(n=1,50, print1(a(n), ", ")) \\ G. C. Greubel, Jul 30 2018

A276531 a(n) = (a(n-1) * a(n-5) + a(n-2) * a(n-3) * a(n-4)) / a(n-6), with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 5, 11, 41, 247, 1498, 39629, 3121233, 1344630757, 4527359876765, 673384475958949877, 12684198948982702826816701, 103442271685605704255863097581658042, 12389248756108266360505757651017660004796444483503, 657084395567781339286109602463271066924826185667810218784212689809097

Offset: 0

Seiichi Manyama, Nov 16 2016

nonn

This sequence is the generalization of Dana Scott's sequence (A048736).
Conjecture: a(n) is an integer for all n. It has been checked by computer for 0 <= n <= 50.
The recursion has the Laurent property. If a(0), ..., a(5) are variables, then a(n) is a Laurent polynomial (a rational function with a monomial denominator). - Michael Somos, Nov 21 2016

Seiichi Manyama, Table of n, a(n) for n = 0..28

Cf. A048736, A006721.

a:=[1,1,1,1,1,1];; for n in [7..25] do a[n]:=(a[n-1]*a[n-5]+a[n-2]*a[n-3]*a[n-4])/a[n-6]; od; a; # Muniru A Asiru, Jul 30 2018

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

RecurrenceTable[{a[n] == (a[n - 1] a[n - 5] + a[n - 2] a[n - 3] a[n - 4])/a[n - 6], a[0] == a[1] == a[2] == a[3] == a[4] == a[5] == 1}, a, {n, 0, 21}] (* Michael De Vlieger, Nov 21 2016 *)
nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,(b*f+d*e*c)/a}; NestList[nxt,{1,1,1,1,1,1},30][[All,1]] (* Harvey P. Dale, Nov 21 2021 *)

def A(k, n)
  a = Array.new(k, 1)
  ary = [1]
  while ary.size < n + 1
    i = a[-1] * a[1] + a[2..-2].inject(:*)
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end
def A276531(n)
  A(6, n)
end

a(n) * a(n-6) = a(n-1) * a(n-5) + a(n-2) * a(n-3) * a(n-4).

a(5-n) = a(n) for all n in Z.

A276532 a(n) = (a(n-1) * a(n-6) + a(n-2) * a(n-3) * a(n-4) * a(n-5)) / a(n-7), with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 11, 41, 371, 7507, 429563, 419408854, 9811194604889, 45615501062085527113, 323645006689468299915979814409, 217332607887523478570092794860281557159140687, 8092345737591989154121803868154457767563221634145658745306515944569

Offset: 0

Seiichi Manyama, Nov 16 2016

nonn

This sequence is one generalization of Dana Scott's sequence (A048736).
a(n) is an integer for all n.
The recursion exhibits the Laurent phenomenon. See A278706 for the exponents of the denominator of the Laurent polynomial. - Michael Somos, Nov 26 2016

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A048736, A006721, A276531, A278706.

def A(k, n)
  a = Array.new(k, 1)
  ary = [1]
  while ary.size < n + 1
    i = a[-1] * a[1] + a[2..-2].inject(:*)
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end
def A276532(n)
  A(7, n)
end

a(n) * a(n-7) = a(n-1) * a(n-6) + a(n-2) * a(n-3) * a(n-4) * a(n-5).

a(6-n) = a(n) for all n in Z.

A360381 Generalized Somos-5 sequence a(n) = (a(n-1)a(n-4) + a(n-2)a(n-3))/a(n-5) = -a(-n), a(1) = 1, a(2) = -1, a(3) = a(4) = 1, a(5) = -7.

Original entry on oeis.org

0, 1, -1, 1, 1, -7, 8, -1, -57, 391, -455, -2729, 22352, -175111, 47767, 8888873, -69739671, 565353361, 3385862936, -195345149609, 1747973613295, -4686154246801, -632038062613231, 34045765616463119, -319807929289790304, -11453004955077020783

Offset: 0

Michael Somos, Feb 04 2023

sign

This has the same recurrence as Somos-5 (A006721) with different initial values.
The elliptic curve y^2 + xy = x^3 + x^2 - 2x (LMFDB label 102.a1) has infinite order point P = (2, 2). The x and y coordinates of n*P have denominators a(n)^2 and |a(n)^3| respectively.
If b(2*n) = 6^(1/4)*a(2*n), b(2*n+1) = a(2*n+1), then b(n) is a generalized Somos-4 sequence with b(n+2)*b(n-2) = 6^(1/2)*b(n+1)*b(n-1) - b(n)*b(n) for all n in Z.
This is the sequence T_n in the Hone 2022 paper.

			5*P = (50/49, 20/343) and a(5) = -7, 6*P = (121/64, -1881/512) and a(6) = 8.

Andrew N. W. Hone, Heron triangles with two rational medians and Somos-5 sequences, European Journal of Mathematics, 8 (2022), 1424-1486; arXiv:2107.03197 [math.NT], 2021-2022.
Andrew N. W. Hone, Heron triangles and the hunt for unicorns, arXiv:2401.05581 [math.NT], 2024.
LMFDB, Elliptic Curve 102.a1 (Cremona label 102a1)

Cf. A006721, A241595.

a[0] = 0; a[1] = a[3] = a[4] = 1; a[2] = -1; a[5] = -7;
a[n_?Negative] := -a[-n];
a[n_] := a[n] = (a[n-1] a[n-4] + a[n-2] a[n-3]) / a[n-5]; (* Andrey Zabolotskiy, Feb 05 2023 *)
a[ n_] := Module[{A = Table[1, Max[5, Abs[n]]]}, A[[2]] = -1; A[[5]] = -7; Do[ A[[k]] = (A[[k-1]]*A[[k-4]] + A[[k-2]]*A[[k-3]])/A[[k-5]], {k, 6, Length[A]}]; If[n==0, 0, Sign[n]*A[[Abs[n]]] ]];

{a(n) = my(A = vector(max(5, abs(n)), k, 1)); A[2] = -1; A[5] = -7; for(k=6, #A, A[k] = (A[k-1]*A[k-4] + A[k-2]*A[k-3])/A[k-5]); if(n==0, 0, sign(n)*A[abs(n)])};

{a(n) = my(E = ellinit([1, 1, 0, -2, 0])); subst(elldivpol(E, n), 'x, 2) *(-1)^(n-1) / 6^((n-1)%2 + n^2\4)}; /* Michael Somos, Mar 01 2025 */

a(2*n) = -A241595(n+1), a(n) = -a(-n) for all n in Z.
From Michael Somos, Aug 19 2025: (Start)
Let S(n) = A006721(n+2) as in Hone. We have for all n in Z:
S(2*n) = S(n-1)*S(n)*a(n-1)*a(n+2) - S(n-2)*S(n+1)*a(n)*a(n+1).
S(2*n+1) = S(n)*S(n+1)*a(n-1)*a(n+2) - S(n-1)*S(n+2)*a(n)*a(n+1).
a(2*n) = a(n)*(a(n-2)*a(n+1)^2 - a(n+2)*a(n-1)^2).
a(2*n+1) = a(n-1)*a(n)^2*a(n+3) - a(n+2)*a(n+1)^2*a(n-2).
S(n-3)*S(n) = S(n-2)*S(n-1) - a(n-2)*a(n-1).
a(n-3)*a(n) = S(n-2)*S(n-1) + a(n-2)*a(n-1).
(End)

A097495 Subsequence of terms of even index in the Somos-5 sequence.

Original entry on oeis.org

1, 1, 1, 3, 11, 83, 1217, 22833, 1249441, 68570323, 11548470571, 2279343327171, 979023970244321, 771025645214210753, 816154448855663209121, 2437052403320731070558403, 7362326966302540624120605547

Offset: 0

Andrew Hone, Aug 24 2004

nonn

The sequence corresponds to the sequence of points Q+nP on the curve y^2 = 4*x^3 - (121/12)*x + 845/216, where Q=(-19/12,2) and P=(17/12,-1).
For every 5th-order bilinear recurrence of Somos-5 type, b(n+3)*b(n-2) = alpha*b(n+2)*b(n-1) + beta*b(n+1)*b(n) (alpha, beta constant), both the subsequence of even index a(n)=b(2n) and that of odd index a(n)=b(2n+1) satisfy the same 4th-order Somos-4 type recurrence a(n+2)*a(n-2) = gamma*a(n+1)*a(n-1) + delta*a(n)^2, where the constant coefficients gamma, delta can be given in terms of alpha, beta and the initial data b(0), b(1), b(2), b(3), b(4).
a(n+2) is the Hankel transform of A174171. - Paul Barry, Mar 10 2010
This is a generalized Somos-4 sequence. - Michael Somos, May 12 2022

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.

Cf. A006721, A006720.

RecurrenceTable[{a[0]==a[1]==a[2]==1,a[3]==3,a[n]==(a[n-1]a[n-3]+ 8a[n-2]^2)/a[n-4]},a,{n,20}] (* Harvey P. Dale, Sep 14 2013 *)
a[ n_] := a[n] = Which[n<1, a[2-n], n<4, {1, 1, 3}[[n]], True, (a[n-1]*a[n-3] + 8*a[n-2]^2)/a[n-4]]; (* Michael Somos, May 12 2022 *)

{a(n) = if(n<1, a(2-n), n<4, [1, 1, 3][n], (a(n-1)*a(n-3) + 8*a(n-2)^2)/a(n-4))}; /* Michael Somos, May 12 2022 */

a(n) = (a(n-1)*a(n-3) + 8*a(n-2)^2)/a(n-4).
Exact formula: a(n) = A*B^n*sigma(c+n*k)/sigma(k)^(n^2) where sigma is the Weierstrass sigma function associated to the elliptic curve y^2 = 4*x^3 - (121/12)*x + 845/216,
   A = 1/sigma(c) = 0.142427718 - 1.037985022*i,
   B = sigma(k)*sigma(c)/sigma(c+k)
     = 0.341936209 + 0.389300717*i,
   c = Integral_{infinity..-19/12} dx/y
     = 0.163392410 + 0.973928783*i,
   k = Integral_{17/12..infinity} dx/y
     = 1.018573545,
all to 9 decimal places.
a(n) = a(2-n) = (-8*a(n-1)*a(n-4) + 57*a(n-2)*a(n-3))/a(n-5) for all n in Z. - Michael Somos, May 12 2022

A242108 a(n) = abs(A242107(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 3, 1, 5, 7, 13, 16, 11, 57, 131, 113, 389, 670, 2311, 3983, 9, 23647, 81511, 140576, 484247, 833503, 5751815, 14871471, 17124617, 147165662, 710017141, 2273917871, 9644648819, 11396432249, 204006839259, 808162720720, 2405317965859

Offset: 0

Michael Somos, Aug 15 2014

nonn

This sequence is similar to Somos-5 (A006721).

Chai Wah Wu, Table of n, a(n) for n = 0..320

Cf. A006721, A242107.

{a(n) = my(v, m); n=abs(n); if( n<6, n>0, v = vector(n, k, 1); for(k=6, n, m = (k+1)%21 - 10; v[k] = ( (-1)^( m%4==0 ) * v[k-1] * v[k-4] + (-1)^( abs((m+4)%8-4)==1 ) * v[k-2] * v[k-3]) / v[k-5]); v[n])};

{a(n) = if( n, sqrtint( denominator( ellmul( ellinit( [1, -1,0, -1, 1]), [0, 1], n)[1])))}; /* Michael Somos, Aug 22 2014 */

from gmpy2 import divexact
A242107 = [0,1,1,1,1,-1]
for n in range(6,321):
    A242107.append(divexact(-A242107[n-1]*A242107[n-4]+
        A242107[n-2]*A242107[n-3],A242107[n-5]))
A242108 = [int(abs(x)) for x in A242107] # Chai Wah Wu, Aug 15 2014

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

cp's OEIS Frontend

A242107 Reduced division polynomials associated with elliptic curve y^2 + x*y = x^3 - x^2 - x + 1 and multiples of point (0, 1).

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A271950 Somos's sequence {b(5,n)} defined in comment in A078495: a(0)=a(1)=...=a(12)=1; for n>=13, a(n)=(a(n-1)*a(n-12)+a(n-6)*a(n-7))/a(n-13).

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A271952 Somos's sequence {b(6,n)} defined in comment in A078495: a(0)=a(1)=...=a(14)=1; for n>=15, a(n)=(a(n-1)*a(n-14)+a(n-7)*a(n-8))/a(n-15).

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A210098 Somos-5 sequence variant: a(n) = (a(n-1) * a(n-4) - a(n-2) * a(n-3)) / a(n-5), a(0) = 0, a(1) = a(2) = a(3) = a(4) = 1, a(5) = 2.

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A271954 Somos's sequence {b(7,n)} defined in comment in A078495: a(0)=a(1)=...=a(16)=1; for n>=17, a(n)=(a(n-1)*a(n-16)+a(n-8)*a(n-9))/a(n-17).

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A276531 a(n) = (a(n-1) * a(n-5) + a(n-2) * a(n-3) * a(n-4)) / a(n-6), with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 1.

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A276532 a(n) = (a(n-1) * a(n-6) + a(n-2) * a(n-3) * a(n-4) * a(n-5)) / a(n-7), with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1.

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A271950 Somos's sequence {b(5,n)} defined in comment in A078495: a(0)=a(1)=...=a(12)=1; for n>=13, a(n)=(a(n-1)a(n-12)+a(n-6)a(n-7))/a(n-13).

A271952 Somos's sequence {b(6,n)} defined in comment in A078495: a(0)=a(1)=...=a(14)=1; for n>=15, a(n)=(a(n-1)a(n-14)+a(n-7)a(n-8))/a(n-15).

A271954 Somos's sequence {b(7,n)} defined in comment in A078495: a(0)=a(1)=...=a(16)=1; for n>=17, a(n)=(a(n-1)a(n-16)+a(n-8)a(n-9))/a(n-17).

A360381 Generalized Somos-5 sequence a(n) = (a(n-1)a(n-4) + a(n-2)a(n-3))/a(n-5) = -a(-n), a(1) = 1, a(2) = -1, a(3) = a(4) = 1, a(5) = -7.