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.

A251792 Decimal expansion of a constant related to A251702.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec and Jon E. Schoenfield, Dec 09 2014

Keywords

Examples

			1.1546796279605837888382808629570944052320556413000593142798453022385779...

Crossrefs

Cf. A086714, A251702.

Programs

  • Magma
    nMax:=160; nExactMax:=20; DP:=100; R:=RealField(DP); SetDefaultRealField(R); logA:=[Log(5.0)]; for n in [2..nMax] do logAprev:=logA[n-1]; if n le nExactMax then Aprev:=Exp(logAprev); logA[n]:=logAprev + Log(Aprev-1) + Log(Aprev-2) - Log(6); else logA[n]:=3*logAprev - Log(6); end if; t:=Exp((1/3^n)*logA[n]); n, ChangePrecision(t,72); end for; // Jon E. Schoenfield, Dec 09 2014
  • Mathematica
    exact = 20; terms = 200; b = ConstantArray[0, terms]; b[[1]] = N[Log[5], 100]; Do[b[[n]] = b[[n - 1]] + If[n > exact, b[[n - 1]], Log[Exp[b[[n - 1]]] - 1]] + If[n > exact, b[[n - 1]], Log[Exp[b[[n - 1]]] - 2]] - Log[6], {n, 2, terms}]; Do[Print[Exp[b[[n]]/3^n]], {n, 1, Length[b]}] (* after Jon E. Schoenfield *)

Formula

Equals lim_{n->infinity} A251702(n)^(1/3^n).

A086714 a(1) = 4, a(n) = a(n-1)*(a(n-1) - 1)/2.

Original entry on oeis.org

4, 6, 15, 105, 5460, 14903070, 111050740260915, 6166133456248548335768188155, 19010600900133834176644234577571914951562754277857057935
Offset: 1

Views

Author

Jon Perry, Jul 29 2003

Keywords

Comments

The next two terms, a(10) and a(11), have 111 and 221 digits. - Harvey P. Dale, Jun 10 2011
Interpretation through plane geometry: Start with the a(n)-sided regular polygon, connect all the vertices to create a figure having a(n+1)=A000217(a(n)-1) edges. Repeat to obtain this sequence. - T. D. Noe, May 13 2016
Let y(1) = x1+x2+x3+x4, and define y(n+1) as the plethysm e2[y(n)], where e2 represents the second elementary symmetric function. Then a(n) is y(n) evaluated at x1=x2=x3=x4=1. - Per W. Alexandersson, Jun 06 2020
Each term is the number of coordinate planes in Euclidean space of the dimensionality of the previous term. - Shel Kaphan, Feb 06 2023

Examples

			a(2) = a(1)*(a(1)-1)/2 = 4*3/2 = 6.

Links

Crossrefs

Cf. A251702, A251794, A006893.
Cf. A000217, A007501, A013589, A050542, A050548, A050536, A050909.

Programs

  • Mathematica
    RecurrenceTable[{a[1]==4,a[n]==(a[n-1](a[n-1]-1))/2},a[n],{n,10}] (* Harvey P. Dale, Jun 10 2011 *)
  • PARI
    v=vector(10,i,(i==1)*4); for(i=2,10,v[i]=v[i-1]*(v[i-1]-1)/2); v
  • PARI
    a086714(upto)={my(a217(n)=n*(n+1)/2,a=4);for(k=1,upto,print1(a,", ");a=a217(a-1))};
a086714(9) \\ Hugo Pfoertner, Sep 18 2021

Formula

Limit_{n->oo} a(n)^(1/2^n) = 1.280497808541657066685323460209089278782... (see A251794). - Vaclav Kotesovec, Feb 15 2014, updated Dec 09 2014
a(n) ~ 2 * A251794^(2^n). - Vaclav Kotesovec, Dec 09 2014
a(n+1) = binomial(a(n), 2). - Shel Kaphan, Feb 06 2023

A129440 a(0)=0, a(1)=1, a(2)=5 and for n>2: a(n) = a(n-1)*(a(n-1) + 1)*(2*a(n-1) + 1)/6.

Original entry on oeis.org

0, 1, 5, 55, 56980, 61667666167030, 78172010815921069181209893626754427513955
Offset: 0

Views

Author

Reinhard Zumkeller, Apr 15 2007

Keywords

Links

Crossrefs

Cf. A000330, A007501, A251702.

Programs

  • Magma
    [0,1] cat [n le 1 select 5 else Self(n-1)*(Self(n-1)+1)*(2*Self(n-1)+1)/6: n in [1..8]]; // G. C. Greubel, Feb 06 2024
  • Mathematica
    Flatten[{0, 1, RecurrenceTable[{a[2] == 5, a[n] == a[n-1]*(a[n-1] + 1)*(2*a[n-1] + 1)/6}, a[n], {n, 8}]}] (* Vaclav Kotesovec, Dec 17 2014 *)
Join[{0,1},NestList[(#(#+1)(2#+1))/6&,5,5]] (* Harvey P. Dale, Sep 13 2022 *)
  • SageMath
    def a(n): # a = A129440
    if n<3: return (0,1,5)[n]
    else: return a(n-1)*(a(n-1)+1)*(2*a(n-1)+1)/6
[a(n) for n in range(9)] # G. C. Greubel, Feb 06 2024

Formula

a(n) = A000330(if n<=2 then n else a(n)).
a(n) ~ sqrt(3) * c^(3^n), where c = 1.13701835838072682283814038264701129587627956851233106833915157... . - Vaclav Kotesovec, Dec 17 2014

A251794 Decimal expansion of a constant related to A086714.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Dec 09 2014

Keywords

Examples

			1.2804978085416570666853234602090892787820401965229548913582461026432...

Crossrefs

Cf. A086714, A251702.

Programs

  • Mathematica
    exact = 32; terms = 200; b = ConstantArray[0, terms]; b[[1]] = N[Log[4], 100]; Do[b[[n]] = b[[n - 1]] + If[n > exact, b[[n - 1]], Log[Exp[b[[n - 1]]] - 1]] - Log[2], {n, 2, terms}]; Do[Print[Exp[b[[n]]/2^n]], {n, 1, Length[b]}] (* after Jon E. Schoenfield *)

Formula

Equals limit n->infinity A086714(n)^(1/2^n).
Showing 1-4 of 4 results.

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