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.

User: Vsevolod F. Lev

Vsevolod F. Lev's wiki page.

Vsevolod F. Lev has authored 2 sequences.

A283438 The largest possible size of a set in F_3^n, free of three-term arithmetic progressions with the difference in {0,1}^n.

Original entry on oeis.org

1, 2, 6, 14, 36
Offset: 0

Views

Author

Vsevolod F. Lev, Mar 07 2017

Keywords

Comments

102 <= a(5) <= 108. - Robert Israel, Mar 10 2017

Examples

			a(0) = 1: {()}.
a(1) = 2: {(0), (1)}.
a(2) = 6: {(0,0), (0,2), (1,1), (1,2), (2,0), (2,1)}.
a(3) = 14: {(0,0,0), (0,0,2), (0,2,0), (0,2,2), (1,0,1), (1,0,2), (1,1,0), (1,1,1), (1,2,0), (1,2,2), (2,0,0), (2,0,1), (2,1,0), (2,1,1)}.

Links

Programs

  • PARI
    \\ This code is for demonstration only, it is far too slow for computation!
ok(v)=vecmax(lift(v))<2
has(v)=my(t); for(i=1,#v, for(j=1,#v, if(i==j,next); for(k=1,#v, if(i==k||j==k,next); t=v[i]-v[j]; if(v[j]-v[k]==t && ok(t), return(0))))); 1
a(n)=my(N=3^n,v=vector(N,i,digits(N+i-1,3)[2..n+1]),u,r,t);for(i=0,2^N-1, u=Mod(vecextract(v,i),3); if(has(u), t=hammingweight(i); if(t>r, r=t))); r \\ Charles R Greathouse IV, Mar 07 2017

Extensions

a(4) from Robert Israel, Mar 07 2017

A003686 Number of genealogical 1-2 rooted trees of height n.

Original entry on oeis.org

1, 2, 3, 5, 11, 41, 371, 13901, 5033531, 69782910161, 351229174914190691, 24509789089655802510792656021, 8608552999157278575508415639286249242844899051
Offset: 1

Views

Author

Vsevolod F. Lev, c. 1998

Keywords

Comments

Let u(n), v(n) be defined by u(1) = v(1) = 1, u(n+1) = u(n) + v(n) and v(n+1) = u(n)*v(n) for n >= 1; then a(n) = u(n) and A064847(n) = v(n). - Benoit Cloitre, Apr 01 2002 [Edited by Petros Hadjicostas, May 11 2020]
Consider the mapping f(a/b) = (a + b)/(a*b). Taking a = 1 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 2/1, 3/2, 5/6, 11/30, ... The current sequence contains the numerators. - Amarnath Murthy, Mar 24 2003
An infinite coprime sequence defined by recursion. - Michael Somos, Mar 19 2004

References

  • D. Parisse, The Tower of Hanoi and the Stern-Brocot Array, Thesis, Munich, 1997.

Links

Crossrefs

Cf. A001622, A001685, A064526, A064847, A070231, A070233, A070234, A094303.

Programs

  • Magma
    I:=[1,2]; [n le 2 select I[n] else Self(n-1)+Self(n-2)*(Self(n-1)-Self(n-2)): n in [1..14]]; // Vincenzo Librandi, Jul 19 2016
  • Mathematica
    RecurrenceTable[{a[1]==1, a[2]==2, a[n]==a[n-1]+a[n-2](a[n-1]-a[n-2])}, a[n],{n,15}] (* Harvey P. Dale, Jul 27 2011 *)
Re[NestList[Re@#+(1+I Re@#)Im@#&, 1+I, 15]] (* Vladimir Reshetnikov, Jul 18 2016 *)
  • PARI
    a(n) = local(an); if(n<1, 0, an=vector(max(2,n)); an[1]=1; an[2]=2; for(k=3, n, an[k]=an[k-1] - an[k-2]^2 + an[k-1]*an[k-2]); an[n])

Formula

Limit_{n -> infinity} a(n)^phi/A064847(n) = 1, where phi = (1 + sqrt(5))/2 is the golden ratio. - Benoit Cloitre, May 08 2002
Numerator of b(n), where b(n) = 1/numerator(b(n-1)) + 1/denominator(b(n-1)) for n >= 2 with b(1) = 1.
a(n+1) = a(n) + a(1)*a(2)*...*a(n-1) for n >= 2. Also a(n+1) = a(n) + a(n-1)*(a(n) - a(n-1)) for n >= 2. In both cases, we start with a(1) = 1 and a(2) = 2.
a(n) ~ c^(phi^n), where c = 1.22508584062304325811405322247537613534139348463831009881946422737141574647... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

Extensions

Additional description from Andreas M. Hinz and Daniele Parisse

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