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.

A156163 Decimal expansion of (19+6*sqrt(2))/17.

Original entry on oeis.org

1, 6, 1, 6, 7, 8, 1, 2, 5, 7, 3, 0, 8, 1, 5, 1, 1, 9, 3, 6, 9, 4, 7, 1, 3, 6, 6, 7, 3, 6, 8, 1, 2, 8, 7, 3, 3, 6, 1, 2, 8, 2, 5, 3, 6, 7, 7, 8, 0, 0, 9, 9, 3, 1, 9, 9, 4, 4, 7, 1, 0, 4, 9, 5, 7, 6, 1, 4, 3, 4, 9, 9, 2, 3, 9, 8, 3, 9, 0, 7, 1, 9, 5, 9, 4, 2, 5, 4, 4, 2, 3, 8, 8, 0, 3, 4, 4, 0, 8, 4, 4, 9, 4, 7, 1
Offset: 1

Views

Author

Klaus Brockhaus, Feb 09 2009

Keywords

Comments

lim_{n -> infinity} b(n)/b(n-1) = (19+6*sqrt(2))/17 for n mod 3 = {0, 2}, b = A155923.
lim_{n -> infinity} b(n)/b(n-1) = ((19+6*sqrt(2))/17)^2 for n mod 3 = {0, 2}, b = A156159.

Examples

			(19+6*sqrt(2))/17 = 1.61678125730815119369...

Links

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)).

Programs

A156160 a(n) = 34*a(n-1)-a(n-2)-2312 for n > 2; a(1)=169, a(2)=2809.

Original entry on oeis.org

169, 2809, 93025, 3157729, 107267449, 3643933225, 123786459889, 4205095700689, 142849467361225, 4852676794578649, 164848161548310529, 5599984815847977025, 190234635577282906009, 6462377624811770824969
Offset: 1

Views

Author

Klaus Brockhaus, Feb 09 2009

Keywords

Comments

lim_{n -> infinity} a(n)/a(n-1) = (17+12*sqrt(2)).

Examples

			a(3) = 34*a(2)-a(1)-2312 = 34*2809-169-2312 = 93025.

Links

Crossrefs

First trisection of A156159.
Cf. A156164 (decimal expansion of (17+12*sqrt(2))).

Programs

  • Mathematica
    LinearRecurrence[{35,-35,1},{169,2809,93025},20] (* Harvey P. Dale, Nov 15 2014 *)
  • PARI
    {m=14; v=concat([169, 2809], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-2312); v}

Formula

a(n) = (578+ (2211-1550*sqrt(2))*(17+12*sqrt(2))^n+(2211+1550*sqrt(2))*(17-12*sqrt(2))^n)/8.
G.f.: x*(169-3106*x+625*x^2)/((1-x)*(1-34*x+x^2)).

Extensions

G.f. corrected by Klaus Brockhaus, Sep 23 2009

A156161 a(n) = 34*a(n-1)-a(n-2)-2312 for n > 2; a(1)=289, a(2)=7225.

Original entry on oeis.org

289, 7225, 243049, 8254129, 280395025, 9525174409, 323575532569, 10992042930625, 373405884106369, 12684808016683609, 430910066683134025, 14638257459209870929, 497269843546452475249, 16892536423120174285225
Offset: 1

Views

Author

Klaus Brockhaus, Feb 09 2009

Keywords

Comments

lim_{n -> infinity} a(n)/a(n-1) = (17+12*sqrt(2)).

Examples

			a(3) = 34*a(2)-a(1)-2312 = 34*7225-289-2312 = 243049.

Links

Crossrefs

Second trisection of A156159.
Equals 289*A008844. - Klaus Brockhaus, Sep 23 2009
Cf. A156164 (decimal expansion of (17+12*sqrt(2))).

Programs

  • Mathematica
    RecurrenceTable[{a[1]==289,a[2]==7225,a[n]==34a[n-1]-a[n-2]-2312},a,{n,20}] (* or *) LinearRecurrence[{35,-35,1},{289,7225,243049},20] (* Harvey P. Dale, Dec 11 2013 *)
  • PARI
    {m=14; v=concat([289, 7225], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-2312); v}

Formula

a(n) = (578+(867-578*sqrt(2))*(17+12*sqrt(2))^n+(867+578*sqrt(2))*(17-12*sqrt(2))^n)/8.
G.f.: x*(289-2890*x+289*x^2)/((1-x)*(1-34*x+x^2)). [corrected by Klaus Brockhaus, Sep 23 2009]
a(1)=289, a(2)=7225, a(3)=243049, a(n) = 35*a(n-1)-35*a(n-2)+a(n-3). - Harvey P. Dale, Dec 11 2013

A156162 a(n) = 34*a(n-1)-a(n-2)-2312 for n > 2; a(1)=625, a(2)=18769.

Original entry on oeis.org

625, 18769, 635209, 21576025, 732947329, 24898630849, 845820499225, 28732998340489, 976076123075089, 33157855186210225, 1126391000208070249, 38264136151888175929, 1299854238163989909025, 44156779961423768728609
Offset: 1

Views

Author

Klaus Brockhaus, Feb 09 2009

Keywords

Comments

lim_{n -> infinity} a(n)/a(n-1) = (17+12*sqrt(2)).

Examples

			a(3) = 34*a(2)-a(1)-2312 = 34*18769-625-2312 = 635209.

Links

Crossrefs

Third trisection of A156159.
Cf. A156164 (decimal expansion of (17+12*sqrt(2))).

Programs

  • Mathematica
    RecurrenceTable[{a[1]==625,a[2]==18769,a[n]==34a[n-1]-a[n-2]-2312},a,{n,20}] (* or *) LinearRecurrence[{35,-35,1},{625,18769,635209},20] (* Harvey P. Dale, Sep 29 2016 *)
  • PARI
    {m=14; v=concat([625 ,18769], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-2312); v}

Formula

a(n) = (578+(387-182*sqrt(2))*(17+12*sqrt(2))^n+(387+182*sqrt(2))*(17-12*sqrt(2))^n)/8.
G.f.: x*(625-3106*x+169*x^2)/((1-x)*(1-34*x+x^2)).

Extensions

G.f. corrected by Klaus Brockhaus, Sep 23 2009
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.