cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Michael Somos, Aug 15 2014

Keywords

Comments

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

Links

Crossrefs

Cf. A006721, A242107.

Programs

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

Formula

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

A244373 a(n) = A242107(n+1) * A242107(n-1) * (1 + mod(n,2)).

Original entry on oeis.org

1, 0, 1, 2, -1, 4, 3, 4, 15, -14, 65, 224, -143, 1824, 1441, 12882, 50959, -151420, 898979, 5337220, 20799, 188372002, -733599, 6648401344, 39471457217, -234341035456, 2785299158305, 24790831385826, 98497628929855, 4377139749257604, -12158771603059997
Offset: 0

Views

Author

Michael Somos, Aug 22 2014

Keywords

Links

Crossrefs

Cf. A242107.

Programs

  • Magma
    I:=[1,2,-1,4,3,4,15]; [n le 7 select I[n] else (-Self(n-6)*Self(n -1) + 2*Self(n-2)*Self(n-5) + 2*Self(n-3)*Self(n-4))/Self(n-7): n in [1..30]]; // G. C. Greubel, Aug 05 2018
  • Mathematica
    Join[{1, 0}, RecurrenceTable[{a[n] == (-a[n-6]*a[n-1] + 2*a[n-2]*a[n-5] + 2*a[n-3]*a[n-4])/a[n-7], a[2] == 1, a[3] == 2, a[4] == -1, a[5] == 4, a[6] == 3, a[7] == 4, a[8] == 15}, a, {n, 2, 50}]] (* G. C. Greubel, Aug 05 2018 *)
  • PARI
    {a(n) = if( n==0, 1, n=abs(n); numerator( ellmul( ellinit([1, -1, 0, -1, 1]), [0, 1], n)[1]))};

Formula

Given elliptic curve "58a1" : y^2 + x * y = x^3 - x^2 - x + 1, then the n th multiple of point [0, 1] is [a(n) / A242107(n)^2, A242107(n+2)^2 * A242107(n-4) / A242107(n)^3].
a(n) = a(-n) for all n in Z.
a(n+1) * A242107(n+4) = a(n+3) * A242107(n) for all n in Z.
0 = a(n)*a(n+7) + a(n+1)*a(n+6) - 2*a(n+2)*a(n+5) - 2*a(n+3)*a(n+4) for all n in Z.
0 = 2*a(n)*a(n+6) - a(n+1)*a(n+5) + 2*a(n+2)*a(n+4) - a(n+3)*a(n+3) for all even n in Z.
0 = a(n)*a(n+6) - 2*a(n+1)*a(n+5) + a(n+2)*a(n+4) - 2*a(n+3)*a(n+3) for all odd n in Z.

A328380 a(n) = (a(n-1) * a(n-3) - 2 * a(n-2)^2) / a(n-4) with a(0) = a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 1, -1, -3, -5, -13, 11, 131, 389, 2311, 9, -81511, -484247, -5751815, -17124617, 710017141, 9644648819, 204006839259, 2405317965859, -84560118880501, -2988387877551859, -105333970856330737, -3722531175803860975, 130866937507290027313
Offset: 0

Views

Author

Michael Somos, Feb 23 2020

Keywords

Comments

This is a (-1,2) generalized Somos-4 sequence.

Links

Crossrefs

Cf. A006720, A242107.

Programs

  • Mathematica
    a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = (a[n - 1]*a[n - 3] - 2*a[n - 2]^2)/a[n - 4]; Array[a, 26, 0] (* Amiram Eldar, Jul 06 2020 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,(d*b-2c^2)/a}; NestList[nxt,{1,1,1,1},30][[;;,1]] (* Harvey P. Dale, Aug 15 2025 *)
  • PARI
    {a(n) = my(v); if( n<0, n=3-n); n++; v = vector(max(4, n), k, 1); for(k=5, n, v[k] = (v[k-1] * v[k-3] - 2*v[k-2]^2) / v[k-4]); v[n]};
  • PARI
    {a(n) = my(m=2*n-3, E=ellinit([1, -1, 0, -1, 1]), z=ellpointtoz(E, [0, 1])); (-1)^n * round(ellsigma(E, m*z) / (ellsigma(E, z)^m^2 * 2^(n^2-3*n+2)))}; /* Michael Somos, Feb 25 2020 */

Formula

a(n) = a(3-n) for all n in Z.
0 = a(n)*a(n+4) - a(n+1)*a(n+3) + 2*a(n+2)*a(n+2) for all n in Z.
0 = a(n)*a(n+5) - 2*a(n+1)*a(n+4) + a(n+2)*a(n+3) for all n in Z.
0 = a(n)*a(n+6) - 4*a(n+1)*a(n+5) - 7*a(n+3)*a(n+3) for all n in Z.
0 = a(n)*a(n+7) - a(n+1)*a(n+6) - 8*a(n+3)*a(n+4) for all n in Z.
A242107(2*n-3) = a(n) for all n in Z.

A178622 A (1, -2) Somos-4 sequence associated to the elliptic curve E: y^2 - 3*x*y - y = x^3 - x.

Original entry on oeis.org

0, 1, 1, 2, 1, -7, -16, -57, -113, 670, 3983, 23647, 140576, -833503, -14871471, -147165662, -2273917871, 11396432249, 808162720720, 14252325989831, 503020937289311, 23268424032702, -625775582778294689, -22086170583356766977, -1557994930804790259136, -27620103680757212617727, 6783061219100782906098017, 547569584492952570186575810
Offset: 0

Views

Author

Paul Barry, May 31 2010

Keywords

Comments

a(n) is (-1)^C(n,2) times the Hankel transform of the sequence with g.f. 1/(1-x^2/(1+2x^2/(1+(1/4)x^2/(1-14x^2/(1-(16/49)x^2/(1-... where 0/1, -2/1, -1/4, 14/1, 16/49, ... are the x-coordinates of the multiples of z=(0, 0) on E.
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 2, z = 1. - Michael Somos, Aug 06 2014

Examples

			G.f. = x + x^2 + 2*x^3 + x^4 - 7*x^5 - 16*x^6 - 57*x^7 - ... - _Michael Somos_, Oct 22 2024

Links

Crossrefs

Cf. A242107.

Programs

  • Magma
    I:=[0, 1, 1, 2, 1]; [n le 5 select I[n] else (Self(n-1)*Self(n-3)-2*Self(n-2)^2)/Self(n-4): n in [1..30]]; // Vincenzo Librandi, Aug 07 2014
  • Mathematica
    nxt[{a_,b_,c_,d_}]:={b,c,d,(d*b-2c^2)/a}; Join[{0},Transpose[ NestList[ nxt,{1,1,2,1},30]][[1]]] (* Harvey P. Dale, Aug 19 2015 *)
Join[{0}, RecurrenceTable[{a[n] == (a[n-1]*a[n-3] -2*a[n-2]^2)/a[n - 4], a[1] == 1, a[2] == 1, a[3] == 2, a[4] == 1}, a, {n, 1, 30}]] (* G. C. Greubel, Sep 18 2018 *)
a[ n_] := Which[n == 0, 0, n < 0, -a[-n], n < 5, {1, 1, 2, 1}[[n]], True, a[n] = (a[n-1]*a[n-3] - 2*a[n-2]*a[n-2])/a[n-4]]; (* Michael Somos, Oct 22 2024 *)
  • PARI
    {a(n) = my(E,z); E=ellinit([3, 0, 1, -1, 0]); z=ellpointtoz(E,[0,0]); -(-1)^n*round(ellsigma(E,n*z)/ellsigma(E,z)^(n^2))};
  • PARI
    m=30; v=concat([0,1,1,2,1], vector(m-5)); for(n=6, m, v[n] = ( v[n-1]*v[n-3] - 2*v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Sep 18 2018

Formula

a(n) = (a(n-1)*a(n-3) - 2*a(n-2)^2)/a(n-4), n>4.
a(n) = -a(-n), a(n+5)*a(n) = 2*a(n+4)*a(n+1) - a(n+3)*a(n+2) for all n in Z. - Michael Somos, Aug 06 2014
a(n) = A242107(2*n) for all n in Z. - Michael Somos, Oct 22 2024

Extensions

Added missing a(0)=0.
More terms from Vincenzo Librandi, Aug 07 2014
Showing 1-4 of 4 results.

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