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.

A010000 a(0) = 1, a(n) = n^2 + 2 for n > 0.

Original entry on oeis.org

1, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Least k such that A070864(k) = 2n-1. - Robert G. Wilson v and Benoit Cloitre, May 20 2002
With an offset of 3, beginning with 6 (deleting first two terms) n*(n+a(n)) + 1 is a cube = (n+1)^3: 1*(1+6) + 1 = 8, 2*(2+11) + 1 = 27, etc. - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003
For n >= 2, a(n) is the maximum element in the continued fraction for Sum_{k>=1} 1/n^(2^k) (for n=2 see A006464). - Benoit Cloitre, Jun 12 2007
Equals binomial transform of [1, 2, 1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 23 2008
Minimum Wiener index of 3-degenerate graphs with n+2 vertices. A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices. The extremal graphs are maximal 3-degenerate graphs with diameter at most 2. - Allan Bickle, Oct 14 2022
a(n-1) is the number of unit triangles enclosed by the triangular spiral drawn on a isometric grid of which the n-th side has length n. The picture in the link shows how the spiral is constructed. - Bob Andriesse, Feb 14 2023

Examples

			G.f. = 1 + 3*x + 6*x^2 + 11*x^3 + 18*x^4 + 27*x^5 + 38*x^6 + 51*x^7 + 66*x^8 + ...

Links

Crossrefs

Cf. A070864. Apart from initial terms, same as A059100.
Cf. A206399.
Cf. A002061 (minimum Wiener index of 2-degenerate graphs).

Programs

  • Mathematica
    a[1] = a[2] = 1; a[n_] := a[n] = 2 + a[n - a[n - 1]]; b = Table[0, {100}]; Do[c = (a[n] + 1)/2; If[c < 101 && b[[c]] == 0, b[[c]] = n], {n, 1, 10^4}]; b
Join[{1}, Range[50]^2 + 2] (* Bruno Berselli, Feb 08 2012 *)
a[ n_] := n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 05 2015 *)
  • Maxima
    a(n):=if n=0 then 1 else 2*n*sum((-1)^(n-j)*binomial(n,j)*(j+1/n)^(n+1),j,0,n)/(n+1)!; /* Vladimir Kruchinin, Jun 03 2013 */
  • PARI
    {a(n) = n^2 + 2 - (n==0)}; /* Michael Somos, Aug 11 2009 */

Formula

a(n) = A000217(n-2) + A000217(n+1) for n > 0. - Jon Perry, Jul 23 2003
Euler transform of length 6 sequence [ 3, 0, 1, 0, 0, -1]. - Michael Somos, Aug 11 2009
From Michael Somos, Aug 11 2009: (Start)
G.f.: (1 + x^3) / (1 - x)^3.
a(n) = a(-n) for all n in Z. (End)
E.g.f.: (x*(x+1) + 2)*e^x - 1. - Gopinath A. R., Feb 14 2012
a(n) = (2*n/(n+1)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(j+1/n)^(n+1), n > 0, a(0)=1. - Vladimir Kruchinin, Jun 03 2013
Sum_{n>=0} 1/a(n) = 3/4+1/4*sqrt(2)*Pi*coth(Pi*sqrt 2)= 1.8610281... - R. J. Mathar, May 07 2024

A078585 Decimal expansion of Sum_{n>=0} 1/4^(2^n).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Dec 01 2002

Keywords

Examples

			0.316421509021893143708079737530525221703311375920552804341210903843055...

Links

Crossrefs

Continued fraction is given in A006464.
Similar sums: A007404, A078885, A078886, A078887, A078888, A078889, A078890, A036987.

Programs

  • Mathematica
    RealDigits[ N[ Sum[1/4^(2^n), {n, 0, Infinity}], 110]][[1]]
  • PARI
    { default(realprecision, 20080); x=suminf(n=0, 1/4^(2^n)); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b078585.txt", n, " ", d)); } \\ Harry J. Smith, May 11 2009

Formula

Equals -Sum_{k>=1} mu(2*k)/(4^k - 1), where mu is the Möbius function (A008683). - Amiram Eldar, Jul 12 2020
Equals A007404 - 1/2. - Kevin Ryde, Nov 11 2020

A061678 Continued fraction for Sum_{n>=0} 1/3^(3^n).

Original entry on oeis.org

0, 2, 1, 2, 3, 26, 1, 2, 2, 1, 2, 19682, 1, 1, 1, 2, 2, 1, 26, 3, 2, 1, 2, 7625597484986, 1, 1, 1, 2, 3, 26, 1, 2, 2, 1, 1, 1, 19682, 2, 1, 2, 2, 1, 26, 3, 2, 1, 2, 443426488243037769948249630619149892802, 1, 1, 1, 2, 3, 26, 1, 2, 2, 1, 2, 19682
Offset: 0

Views

Author

Jason Earls, Jun 23 2001

Keywords

Comments

The continued fraction has a "folded" overall structure. Apart from a(0) and from the record values of the form 3^(3^k)-1 (k >= 0), the only terms are 1 and 3. This follows from the theorem in Shallit's paper. - Georg Fischer, Aug 29 2022

Examples

			0.370421175633926798495743189411...

Links

Crossrefs

Cf. A004200, A006464, A006465, A006466, A006467, A007400, A081771.

Programs

  • Mathematica
    ContinuedFraction[Sum[1/3^(3^i), {i, 0, 5}]] (* Michael De Vlieger, Jul 01 2018 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 8000); x=contfrac(suminf(n=0, 1/3^(3^n))); for (n=0, 382, write("b061678.txt", n, " ", x[n+1])) } \\ Harry J. Smith, Jul 26 2009

A081772 Continued fraction for sum(k>=0,1/4^(2^k-1)).

Original entry on oeis.org

1, 3, 1, 3, 4, 3, 1, 3, 3, 1, 3, 3, 1, 2, 1, 3, 3, 1, 3, 4, 3, 1, 3, 3, 1, 2, 1, 3, 4, 3, 1, 3, 3, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 4, 3, 1, 3, 3, 1, 2, 1, 3, 4, 3, 1, 3, 3, 1, 3, 3, 1, 2, 1, 3, 3, 1, 3, 4, 3, 1, 2, 1, 3, 3, 1, 3, 4, 3, 1, 3, 3, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 4, 3, 1, 3, 3, 1, 2, 1, 3, 4, 3, 1, 3, 3
Offset: 1

Views

Author

Benoit Cloitre, Apr 10 2003

Keywords

Comments

Contains only elements 1<=a(n)<=4

Crossrefs

Cf. A004200, A006464.
Showing 1-4 of 4 results.

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

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