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 1-3 of 3 results.

A110654 a(n) = ceiling(n/2), or: a(2*k) = k, a(2*k+1) = k+1.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 05 2005

Keywords

Comments

The number of partitions of 2n into exactly 2 odd parts. - Wesley Ivan Hurt, Jun 01 2013
Number of nonisomorphic outer planar graphs of order n >= 3 and size n+1. - Christian Barrientos and Sarah Minion, Feb 27 2018
Also the clique covering number of the n-dipyramidal graph for n >= 3. - Eric W. Weisstein, Jun 27 2018

Examples

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 5*x^9 + ...

Links

Crossrefs

Essentially the same sequence as A008619 and A123108.
Cf. A014557, A275416 (multisets).
Cf. A298648 (number of smallest coverings of dipyramidal graphs by maximal cliques).

Programs

Formula

a(n) = floor(n/2) + n mod 2.
a(n) = A004526(n+1) = A001057(n)*(-1)^(n+1).
For n > 0: a(n) = A008619(n-1).
A110655(n) = a(a(n)), A110656(n) = a(a(a(n))).
a(n) = A109613(n) - A028242(n) = A110660(n) / A028242(n).
a(n) = A001222(A029744(n)). - Reinhard Zumkeller, Feb 16 2006
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 2, a(2) = a(1) = 1, a(0) = 0. - Reinhard Zumkeller, May 22 2006
First differences of quarter-squares: a(n) = A002620(n+1) - A002620(n). - Reinhard Zumkeller, Aug 06 2009
a(n) = A007742(n) - A173511(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = A000217(n) / A008619(n). - Reinhard Zumkeller, Aug 24 2011
From Michael Somos, Sep 19 2006: (Start)
Euler transform of length 2 sequence [1, 1].
G.f.: x/((1-x)*(1-x^2)).
a(-1-n) = -a(n). (End)
a(n) = floor((n+1)/2) = |Sum_{m=1..n} Sum_{k=1..m} (-1)^k|, where |x| is the absolute value of x. - William A. Tedeschi, Mar 21 2008
a(n) = A065033(n) for n > 0. - R. J. Mathar, Aug 18 2008
a(n) = ceiling(n/2) = smallest integer >= n/2. - M. F. Hasler, Nov 17 2008
If n is zero then a(n) is zero, else a(n) = a(n-1) + (n mod 2). - R. J. Cano, Jun 15 2014
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + x) * u * v - (u^2 - v) / 2. - Michael Somos, Jun 15 2014
Given g.f. A(x) then 2 * x^3 * (1 + x) * A(x) * A(x^2) is the g.f. of A014557. - Michael Somos, Jun 15 2014
a(n) = (n + (n mod 2)) / 2. - Fred Daniel Kline, Jun 08 2016
E.g.f.: (sinh(x) + x*exp(x))/2. - Ilya Gutkovskiy, Jun 08 2016
Satisfies the nested recurrence a(n) = a(a(n-2)) + a(n-a(n-1)) with a(1) = a(2) = 1. Cf. A004001. - Peter Bala, Aug 30 2022

Extensions

Deleted wrong formula and added formula. - M. F. Hasler, Nov 17 2008

A110657 a(n) = A028242(A028242(n)).

Original entry on oeis.org

0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 7, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 13, 14, 12, 13, 14, 15, 13, 14, 15, 16, 14, 15, 16, 17, 15, 16, 17, 18, 16, 17, 18, 19, 17, 18, 19, 20, 18, 19, 20, 21, 19, 20, 21
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 05 2005

Keywords

Comments

Also array read by rows, with four columns, in which row n lists n, n+1, n+2, n. - Omar E. Pol, Jan 22 2012

Examples

			From _Omar E. Pol_, Jan 22 2012: (Start)
Array begins:
0, 1, 2, 0;
1, 2, 3, 1;
2, 3, 4, 2;
3, 4, 5, 3;
4, 5, 6, 4;
5, 6, 7, 5;
6, 7, 8, 6;
7, 8, 9, 7;
(End)

Links

Crossrefs

Cf. A007892, A110655.

Programs

  • Magma
    [Integers()!(2*n-6*(-1)^(n*(n+1)/2)+3*(-1)^n+3)/8: n in [0..81]]; // Bruno Berselli, Sep 28 2011
  • Maple
    A110657:=n->(1/8)*(2*n-6*(-1)^(n*(n+1)/2)+3*(-1)^n+3): seq(A110657(n), n=0..100); # Wesley Ivan Hurt, Apr 12 2015
  • Mathematica
    Table[(1/8)*(2*n - 6*(-1)^(n*(n + 1)/2) + 3*(-1)^n + 3), {n, 0, 100}] (* Wesley Ivan Hurt, Apr 12 2015 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,0,1},90] (* Harvey P. Dale, Feb 02 2020 *)
  • PARI
    vector(80, n, n--; 1 + (n-7)\4 + ((n-7) % 4)) \\ Michel Marcus, Apr 13 2015

Formula

A110658(n) = A028242(a(n)) = a(A028242(n)).
a(n) = floor(n/4) + (n mod 4) mod 3.
From Bruno Berselli, Sep 28 2011: (Start)
G.f.: x*(1+x-2*x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2).
a(n) = (1/8)*(2*n-6*(-1)^(n*(n+1)/2)+3*(-1)^n+3). (End)
From Wesley Ivan Hurt, Apr 12 2015: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5).
a(n) = 1 + floor((n-7)/4) + ((n-7) mod 4). (End)
a(n) = n - 3*floor((n+1)/4). - Gionata Neri, Oct 19 2015
a(n) = (2*n+3-6*cos(n*Pi/2)+3*cos(n*Pi)+6*sin(n*Pi/2))/8. - Wesley Ivan Hurt, Oct 01 2017
Sum_{n>=4} (-1)^(n+1)/a(n) = 1/2. - Amiram Eldar, Oct 04 2022

A110656 a(n) = A110654(A110654(A110654(n))).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 05 2005

Keywords

Links

Crossrefs

Cf. A110658.

Programs

  • Mathematica
    Table[Ceiling[n/8], {n, 0, 50}] (* G. C. Greubel, Sep 03 2017 *)
  • PARI
    for(n=0,50, print1(ceil[n/8], ", ")) \\ G. C. Greubel, Sep 03 2017

Formula

a(n) = ceiling(n/8).
a(n) = A110654(A110655(n)) = A110655(A110654(n)).
G.f.: x / ((x-1)^2*(x+1)*(x^2+1)*(x^4+1)). - Colin Barker, Apr 01 2013
Showing 1-3 of 3 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.