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

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

Author

Reinhard Zumkeller, Aug 05 2005

Keywords

• nonn
• easy

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

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Clique Covering Number
• Eric Weisstein's World of Mathematics, Dipyramidal Graph
• Wikipedia, Floor and ceiling functions
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

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

Haskell

a110654 = (`div` 2) . (+ 1)
a110654_list = tail a004526_list  -- Reinhard Zumkeller, Jul 27 2012

Magma

[Ceiling(n/2): n in [0..80]]; // Vincenzo Librandi, Nov 04 2014

Mathematica

a[ n_] := Ceiling[ n / 2]; (* Michael Somos, Jun 15 2014 *)
a[ n_] := Quotient[ n, 2, -1]; (* Michael Somos, Jun 15 2014 *)
a[0] = 0; a[n_] := a[n] = n - a[n - 1]; Table[a[n], {n, 0, 100}] (* Carlos Eduardo Olivieri, Dec 22 2014 *)
CoefficientList[Series[x^/(1 - x - x^2 + x^3), {x, 0, 75}], x] (* Robert G. Wilson v, Feb 05 2015 *)
LinearRecurrence[{1, 1, -1}, {0, 1, 1}, 75] (* Robert G. Wilson v, Feb 05 2015 *)

PARI

a(n)=n\2+n%2;

PARI

a(n)=(n+1)\2; \\ M. F. Hasler, Nov 17 2008

Python

def A110654(n): return n+1>>1 # Chai Wah Wu, Jun 27 2025

Sage

[floor(n/2) + 1 for n in range(-1,75)] # Zerinvary Lajos, Dec 01 2009

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