A275416 -id:A275416 - cp's OEIS frontend

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

Reinhard Zumkeller, Aug 05 2005

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

			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 + ...

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).

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

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

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

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;
```

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

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

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

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

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

A269628 Dimension of BSym_n.

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 35, 57, 102, 165, 279, 444, 726, 1136, 1804, 2785, 4326, 6584, 10048, 15100, 22698, 33723, 50034, 73557, 107912, 157122, 228189, 329341, 473998, 678576, 968672, 1376402, 1950177, 2751900, 3872346, 5429166, 7591294, 10579486, 14705595, 20379419, 28172006, 38836332, 53410265, 73264431, 100271052

Offset: 0

Eric S. Egge, Mar 01 2016

nonn

BSym_n is the space of homogeneous series of degree n in the variables x_1, x_{-1}, x_2, x_{-2}, ... which are invariant under the natural action of the hyperoctahedral group.
a(n) is also the number of Ferrers diagrams (in the English convention) in which some boxes contain a dot, such that the dots are left-justified in each row, and for each k, the dots in rows with k boxes form a Ferrers shape, and there are n total dots and boxes.
a(n) is also the number of partitions of n in which there are 1 + floor(k/2) different parts of "type" k for each k.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Vaclav Kotesovec)

Cf. A275416.

a:= proc(n) option remember; `if`(n=0, 1, add(add(d*ceil(
      (d+1)/2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Sep 20 2017

Table[SeriesCoefficient[1/Product[(1 - x^j)^Floor[(j + 2)/2], {j, 1, n}], {x, 0, n}], {n, 0, 44}] (* Michael De Vlieger, Mar 06 2016 *)

N=66;  x='x+O('x^N); Vec( 1 / prod(j=1, N, (1-x^j)^floor((j+2)/2) ) ) \\ Joerg Arndt, Mar 02 2016

G.f.: 1 / (Product_{j>=1} (1-x^j)^floor((j+2)/2)).
a(n) ~ Zeta(3)^(25/72) / (A^(1/2) * Pi * 2^(3/4) * sqrt(3) * n^(61/72)) * exp(1/24 - Pi^4/(384*Zeta(3)) + Pi^2*n^(1/3) / (8*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 06 2016
a(n) = A275416(2n,n). - Alois P. Heinz, Sep 19 2017

cp's OEIS Frontend

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

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A269628 Dimension of BSym_n.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

Python

Sage

Formula

Extensions

A269628 Dimension of BSym_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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