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.

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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).

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

Views

Author

Robert G. Wilson v, Nov 20 2000

Keywords

References

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

Links

Crossrefs

Cf. A006720, A006721, A006722 and A006723.

Programs

  • Magma
    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
  • Mathematica
    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}]
  • PARI
    {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

A058232 a(n) = (a(n-1)a(n-5) + a(n-2)a(n-4) + a(n-3)^2)/a(n-6).

Original entry on oeis.org

0, 1, 0, 1, 1, -1, -1, 0, 0, 1, -1, -1, -1, -2, 1, 2, -1, 2, 1, -3, -3, -1, -4, 4, 1, -3, -5, -9, 8, 15, -4, 17, -8, -23, -3, -21, -49, 52, 76, -47, 11, -133, 79, 238, 97, 518, -417, -750, 625, -647, -343, 1967, 3048, -1000, 5553, -8375, -4233, 13375, 10912, 33503
Offset: 0

Views

Author

Michael Somos, Dec 01 2000

Keywords

Comments

Satisfies the defining recursion for the Somos-6 sequence. - Michael Somos, May 25 2014

References

  • N. D. Elkies, email, Nov 29 2000.

Links

Crossrefs

Cf. A006722.

Programs

  • Mathematica
    nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,(f*b+e*c+d^2)/a}; Join[ {0,1,0,1,1,-1,-1,0,0}, Transpose[ NestList[ nxt,{1,-1,-1,-1,-2,1},50]][[1]]] (* Harvey P. Dale, Apr 06 2013 *)
  • PARI
    {a(n) = local(an, a0, num); if( n<0, -a(-n), if( n==0, 0, a0 = [1, 0, 1, 1, -1, -1, 0, 0, 1, -1, -1, -1, -2, 1]; an = vector(n); for( k=1, n, an[k] = if( k<15, a0[k], (num = an[k-1] * an[k-5] + an[k-2] * an[k-4] + an[k-3]^2) / an[k-6])); an[n]))};

Formula

a(-n) = -a(n). a(n+6) * a(n-6) = a(n+4) * a(n-4) + a(n+2) * a(n-2) for all n in Z.
a(n+6) * a(n-6) = -a(n+5) * a(n-5) + 2*a(n+4) * a(n-4) - a(n)^2 for all n in Z. - Michael Somos, May 25 2014
a(n+6) * a(n-5) = - a(n+4) * a(n-3) + a(n+2) * a(n-1) for all n in Z. - Michael Somos, May 25 2014
a(n+5) * a(n-4) = a(n+4) * a(n-3) + a(n+3) * a(n-2) - a(n+2) * a(n-1) + a(n+1) * a(n) for all n in Z. - Michael Somos, May 25 2014

A122025 a(n) = (3*a(n-1)*a(n-4) - a(n-2)*a(n-3)) / a(n-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 13, 29, 109, 629, 4274, 23329, 170353, 2034385, 35152177, 494532481, 7768108946, 176258003381, 7248370100941, 266967020984381, 9997125891977197, 431397731193035525, 38079906742644329378
Offset: 0

Views

Author

Roger L. Bagula, Sep 13 2006

Keywords

Crossrefs

Cf. A006722.

Programs

  • Mathematica
    a[n_] := a[n] = (3*a[n - 1]a[n - 4] - a[n - 2]*a[n - 3])/a[n - 5]; a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1; a[4] = 1; Table[a[n], {n, 0, 30}]
RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==a[5]==1,a[n]==(3a[n-1]a[n-4]-a[n-2]a[n-3])/a[n-5]},a,{n,30}] (* Harvey P. Dale, Sep 02 2017 *)

Extensions

Edited by N. J. A. Sloane, Sep 17 2006

A227999 a(n) = (a(n-1) * a(n-5) + a(n-2) * a(n-4) + a(n-3)^2) / a(n-6), a(0) = a(1) = a(2) = a(3) = 1, a(4) = a(5) = 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 11, 25, 97, 220, 1396, 6053, 30467, 249431, 1381913, 19850884, 160799404, 1942868797, 36133524445, 458473480079, 13521902050025, 220176552243482, 7006033824529130, 276364333237297549, 7470025110120086101, 460097285931623600317, 17010560092754291510533, 1372227474279446678113082
Offset: 0

Views

Author

Max Alekseyev, Dec 04 2013

Keywords

Comments

David Speyer showed (modulo some empirical relations) that all terms are integer.

Links

Crossrefs

A variation of A006722.

Programs

  • Magma
    I:=[1,1,1,1,2,2]; [n le 6 select I[n] else (Self(n-1)*Self(n-5) + Self(n-2)*Self(n-4) + Self(n-3)^2)/Self(n-6): n in [1..30]]; // G. C. Greubel, Aug 08 2018
  • Mathematica
    RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[4]==a[5]==2,a[n]==(a[n-1] a[n-5]+ a[n-2]a[n-4]+a[n-3]^2)/a[n-6]},a,{n,30}] (* Harvey P. Dale, Nov 11 2014 *)
a[ n_] := Which[ Abs[n - 3/2] < 2, 1, Abs[n - 3/2] < 4, 2, n < 0, a[3 - n], True, (a[n - 1] a[n - 5] + a[n - 2] a[n - 4] + a[n - 3]^2) / a[n - 6]]; (* Michael Somos, Jul 24 2018 *)
  • PARI
    {a(n) = my(v); if( n<2, n = 3-n); n++; v = vector(n, i, 1+(i>4)); for(k=7, n, v[k] = (v[k-1]*v[k-5] + v[k-2]*v[k-4] + v[k-3]*v[k-3]) / v[k-6]); v[n]}; /* Michael Somos, Apr 25 2017 */

Formula

For n>=6, a(n) = (a(n-1) * a(n-5) + a(n-2) * a(n-4) + a(n-3)^2) / a(n-6).
a(n) = a(3-n) for all n in Z. - Michael Somos, Apr 25 2017
0 = a(n)*a(n+9) +a(n+1)*a(n+8) +a(n+2)*a(n+7) -a(n+3)*a(n+6) -32*a(n+4)*a(n+5) for all n in Z. - Michael Somos, Apr 25 2017
0 = a(n)*a(n+10) -a(n+1)*a(n+9) -32*a(n+2)*a(n+8) +17*a(n+3)*a(n+7) +49*a(n+4)*a(n+6) for all n in Z. - Michael Somos, Apr 25 2017
0 = +a(n)*a(n+11) -32*a(n+1)*a(n+10) -33*a(n+2)*a(n+9) -49*a(n+3)*a(n+8) +1007*a(n+5)*a(n+6) for all n in Z. - Michael Somos, Apr 25 2017

A122024 a(n) = (3*a(n - 1)a(n - 5) - a(n - 2)*a(n - 4))/a(n - 6); a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 1, a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 5, 13, 34, 76, 286, 1651, 10933, 76130, 418553, 3071419, 37191253, 597219127, 11052077737, 158194282219, 2500524791374, 60075261409487, 2150997470890553, 103505410541423890, 3957922747010968084
Offset: 0

Views

Author

Roger L. Bagula, Sep 12 2006

Keywords

Crossrefs

Cf. Somos-6 sequence A006722.

Programs

  • Mathematica
    a[n_] := a[n] = (3*a[n - 1]a[n - 5] - a[n - 2]*a[n - 4])/a[n - 6]; a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1; a[4] = 1; a[5] = 1; Table[a[n], {n, 0, 30}]
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==1,a[n]==(3a[n-1] a[n-5]- a[n-2]a[n-4])/a[n-6]},a,{n,30}] (* Harvey P. Dale, Nov 07 2013 *)

Extensions

Edited by N. J. A. Sloane, Sep 13 2006
Offset changed to 0 by Georg Fischer, Jun 18 2021
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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