A035104 -id:A035104 - cp's OEIS frontend

A004652 Expansion of x(1+x^2+x^4)/((1-x)(1-x^2)*(1-x^3)).

0, 1, 1, 3, 4, 7, 9, 13, 16, 21, 25, 31, 36, 43, 49, 57, 64, 73, 81, 91, 100, 111, 121, 133, 144, 157, 169, 183, 196, 211, 225, 241, 256, 273, 289, 307, 324, 343, 361, 381, 400, 421, 441, 463, 484, 507, 529, 553, 576, 601, 625, 651, 676, 703, 729, 757, 784, 813

Offset: 0

N. J. A. Sloane

As a Molien series this arises as (1+x^12)/((1-x^4)*(1-x^8)^2).
Starting (1, 3, 4, ...) = row sums of an infinite triangle with alternate columns of (1, 2, 3, ...) and (1, 0, 0, 0, ...). - Gary W. Adamson, May 14 2010
a(n) is also the number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and one square has one of the colors. See the formula from A054772. - Wolfdieter Lang, Oct 03 2016
Also the genus of the complete bipartite graph K_{n+2,n+2}. - Eric W. Weisstein, Jan 19 2018

			From _Gary W. Adamson_, May 14 2010: (Start)
First few rows of the generating triangle =
1;
2, 1;
3, 0, 1;
4, 0, 2, 1;
5, 0, 3, 0, 1;
6, 0, 4, 0, 2, 1;
7, 0, 5, 0, 3, 0, 1;
8, 0, 6, 0, 4, 0, 2, 1;
...
Example: a(7) = 13 = (6 + 0 + 4 + 0 + 2 + 1). (End)
x + x^2 + 3*x^3 + 4*x^4 + 7*x^5 + 9*x^6 + 13*x^7 + 16*x^8 + 21*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (Abstract, pdf, ps).
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
J. E. Strapasson, S. I. R. Costa, and M. M. S. Alves, On Genus of Circulant Graphs, arXiv:1004.0244 [math.GN], 2010-2016. - _Jonathan Vos Post_, Apr 05 2010
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Graph Genus
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

First differences give A028242. Cf. A035104, A035106.
A002061(n)=a(2*n-1). A035104(n)=a(n+7)-12. A035106(n)=a(n+3)-1.
Cf. A135840, A000290.
Column 1 of A195040. - Omar E. Pol, Sep 28 2011
Cf. A054772, column 2.

a004652 = ceiling . (/ 4) . fromIntegral . (^ 2)
a004652_list = 0 : 1 : zipWith (+) a004652_list [1..]
-- Reinhard Zumkeller, Dec 18 2013

[Ceiling(n^2/4): n in [0..60] ]; // Vincenzo Librandi, Aug 19 2011

with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=2)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,ZL),size=m+3),m=0..57) ; # Zerinvary Lajos, Mar 09 2007

CoefficientList[Series[x (1 - x + x^2)/((1 - x)^2*(1 - x^2)), {x, 0, 57}], x] (* Michael De Vlieger, Oct 03 2016 *)
Table[Ceiling[n^2/4], {n, 0, 20}] (* Eric W. Weisstein, Jan 19 2018 *)
Ceiling[Range[0, 20]^2/4] (* Eric W. Weisstein, Jan 19 2018 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 1, 3, 4}, {0, 20}] (* Eric W. Weisstein, Jan 19 2018 *)

PARI
```
{a(n) = ceil(n^2 / 4)}
```

a(n) = ceiling(n^2/4).
a(-n) = a(n).
G.f.: x * (1 - x + x^2) / ((1 - x)^2 * (1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + 1. a(2*n) = n^2, a(2*n-1) = n^2 - n + 1. - Michael Somos, Apr 21 2000
Interleaves square numbers with centered polygonal numbers: a(2*n)=A000290(n), a(2*n+1)=A002061(n+1). - Paul Barry, Mar 13 2003
For n > 1: a(n) is the digit reversal of n in base A008619(n), where a(n) is written in base 10. - Naohiro Nomoto, Mar 15 2004
a(n) = a(n-2) + n - 1. - Paul Barry, Jul 14 2004
Euler transform of length 6 sequence [ 1, 2, 1, 0, 0, -1]. - Michael Somos, Apr 03 2007
Starting (1, 3, 4, 7, 9, 13, ...), row sums of triangle A135840. - Gary W. Adamson, Dec 01 2007
a(n) = (3/8)*(-1)^(n+1) + 5/8 - (3/4)*(n+1) + (1/4)*(n+2)*(n+1). - Richard Choulet, Nov 27 2008
a(n) = n^2/4 - 3*((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011
a(n) = -n + floor( (n+1)(n+3)/4 ). - Wesley Ivan Hurt, Jun 23 2013
a(n) = A054772(n, 1) = A054772(n, n^2-1), n >= 1. - Wolfdieter Lang, Oct 03 2016
E.g.f.: (x*(x + 1)*exp(x) + 3*sinh(x))/4. - Ilya Gutkovskiy, Oct 03 2016
a(n) = binomial(floor((n+3)/2),2) + binomial(floor((n+(-1)^n)/2),2). - Yuchun Ji, Feb 03 2021

A035106 1, together with numbers of the form k(k+1) or k(k+2), k > 0.

Original entry on oeis.org

1, 2, 3, 6, 8, 12, 15, 20, 24, 30, 35, 42, 48, 56, 63, 72, 80, 90, 99, 110, 120, 132, 143, 156, 168, 182, 195, 210, 224, 240, 255, 272, 288, 306, 323, 342, 360, 380, 399, 420, 440, 462, 483, 506, 528, 552, 575, 600, 624, 650, 675, 702, 728, 756, 783, 812, 840

Offset: 1

N. J. A. Sloane, revised Oct 30 2001

Largest integer m such that every permutation (p_1, ..., p_n) of (1, ..., n) satisfies p_i * p_{i+1} >= m for some i, 1 <= i <= n-1. Equivalently, smallest integer m such that there exists a permutation (p_1, ..., p_n) of (1, ..., n) satisfying p_i * p_{i+1} <= m for every i, 1 <= i <= n-1.
Also, nonsquare positive integers m such that floor(sqrt(m)) divides m. - Max Alekseyev, Nov 27 2006
Also, for n>1, a(n) is the number of non-isomorphic simple connected undirected graphs having n+1 edges and a longest path of length n. - Nathaniel Gregg, Nov 02 2021

			n=5: we must arrange the numbers 1..5 so that the max of the products of pairs of adjacent terms is minimized. The answer is 51324, with max product = 8, so a(5) = 8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A006446, A052938, A064764, A064796, A064797, A064817, A004652, A035104, A035107.
First differences give (essentially) A028242.
Bisections: A002378 (pronic numbers) and A005563.

Concatenation([1], List([2..60], n-> (2*n*(n+2) +3*((-1)^n -1))/8)); # G. C. Greubel, Jun 10 2019

import Data.List.Ordered (union)
a035106 n = a035106_list !! (n-1)
a035106_list = 1 : tail (union a002378_list a005563_list)
-- Reinhard Zumkeller, Oct 05 2015

[1] cat [(2*n*(n+2) +3*((-1)^n -1))/8: n in [2..60]]; // G. C. Greubel, Jun 10 2019

Join[{1},LinearRecurrence[{2,0,-2,1},{2,3,6,8},60]] (* or *) Join[{1}, Table[ If[EvenQ[n],(n(n+2))/4,((n-1)(n+3))/4],{n,2,60}]] (* Harvey P. Dale, May 03 2012 *)

my(x='x+O('x^60)); Vec(x*(x^4-2*x^3+x^2-1)/((x-1)^3*(x+1))) \\ Altug Alkan, Oct 23 2015

A035106(n)=!(n-1)+floor((n^2)/4+n/2); \\ R. J. Cano, Jul 24 2023

[1]+[(2*n*(n+2) +3*((-1)^n -1))/8 for n in (2..60)] # G. C. Greubel, Jun 10 2019

For n > 1, a(n) = n*(n+2)/4 if n is even and (n-1)*(n+3)/4 if n is odd. - Jud McCranie, Oct 25 2001
a(n) = a(n-1) + a(n-2) - a(n-3) + 1 = A002620(n+2) + A004526(n+2). - Henry Bottomley, Mar 08 2000
a(n+2) = (2*n^2 + 12*n + 3*(-1)^n + 13)/8, with a(1)=1, i.e., a(n+2) = (n+2)*(n+4)/4 if n is even and (n+1)*(n+5)/4 if n is odd. - Vladeta Jovovic, Oct 23 2001
From Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004: (Start)
a(n) = a(n-2) + (n-1), where a(1) = 0, a(2) = 0.
a(n) = (2*(n+1)^2 + 3*(-1)^n - 5)/8, n>=2, with a(1)=1. (End)
For n > 1, a(n) = floor((n+1)^4/(4*(n+1)^2+1)). - Gary Detlefs, Feb 11 2010
For n > 1, a(n) = n + ceiling((1/4)*(n-1)^2) - 1. - Clark Kimberling, Jan 07 2011; corrected by Arkadiusz Wesolowski, Sep 25 2012
a(1)=1, a(2)=2, a(3)=3, a(4)=6, a(5)=8; for n > 5, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, May 03 2012
G.f.: x + x^2*(2-x) / ( (1+x)*(1-x)^3 ) = x*(x^4 - 2*x^3 + x^2 - 1)/((x-1)^3*(x+1)). - Vladeta Jovovic, Oct 23 2001; Harvey P. Dale, May 03 2012
a(n) = floor(n/2)*(1 + ceiling(n/2)), a(1) = 1. - Arkadiusz Wesolowski, Sep 25 2012
a(n) = ceiling((n-1)*(n+3)/4), n > 1. - Wesley Ivan Hurt, Jun 14 2013
a(n+1) - a(n) = A052938(n-2) for n > 1. - Reinhard Zumkeller, Oct 06 2015
E.g.f.: (8*x + 3*exp(-x) - (3-6*x-2*x^2)*exp(x))/8. - G. C. Greubel, Jun 10 2019
Sum_{n>=1} 1/a(n) = 11/4. - Amiram Eldar, Sep 24 2022

Edited by Max Alekseyev, Oct 09 2015

Definition modified to allow for the initial 1. - N. J. A. Sloane, May 17 2016

A035107 First differences give (essentially) A028242.

Original entry on oeis.org

3, 9, 17, 29, 44, 64, 88, 118, 153, 195, 243, 299, 362, 434, 514, 604, 703, 813, 933, 1065, 1208, 1364, 1532, 1714, 1909, 2119, 2343, 2583, 2838, 3110, 3398, 3704, 4027, 4369, 4729, 5109, 5508, 5928, 6368, 6830, 7313, 7819, 8347, 8899, 9474

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A028242, A004652, A035104, A035106.

[(4*n^3+54*n^2+212*n+153-9*(-1)^n)/48: n in [0..50]]; // Vincenzo Librandi, Oct 21 2013

LinearRecurrence[{3,-2,-2,3,-1},{3,9,17,29,44},50] (* Harvey P. Dale, Oct 20 2013 *)
CoefficientList[Series[(2 x^3 - 4 x^2 + 3)/((x - 1)^4 (x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 21 2013 *)

a(n) = (4*n^3 +54*n^2 +212*n +153 -9*(-1)^n)/48.

G.f.: (2*x^3-4*x^2+3) / ((x-1)^4*(x+1)). - Colin Barker, Mar 04 2013

cp's OEIS Frontend

A004652 Expansion of x*(1+x^2+x^4)/((1-x)*(1-x^2)*(1-x^3)).

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A035106 1, together with numbers of the form k*(k+1) or k*(k+2), k > 0.

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A035107 First differences give (essentially) A028242.

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A004652 Expansion of x(1+x^2+x^4)/((1-x)(1-x^2)*(1-x^3)).

A035106 1, together with numbers of the form k(k+1) or k(k+2), k > 0.