A007590 -id:A007590 - cp's OEIS frontend

A047838 a(n) = floor(n^2/2) - 1.

1, 3, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611, 647, 683, 721, 759, 799, 839, 881, 923, 967, 1011, 1057, 1103, 1151, 1199, 1249, 1299, 1351, 1403

Offset: 2

Michael Somos, May 07 1999

Define the organization number of a permutation pi_1, pi_2, ..., pi_n to be the following. Start at 1, count the steps to reach 2, then the steps to reach 3, etc. Add them up. Then the maximal value of the organization number of any permutation of [1..n] for n = 0, 1, 2, 3, ... is given by 0, 1, 3, 7, 11, 17, 23, ... (this sequence). This was established by Graham Cormode (graham(AT)research.att.com), Aug 17 2006, see link below, answering a question raised by Tom Young (mcgreg265(AT)msn.com) and Barry Cipra, Aug 15 2006
From Dmitry Kamenetsky, Nov 29 2006: (Start)
This is the length of the longest non-self-intersecting spiral drawn on an n X n grid. E.g., for n=5 the spiral has length 17:
  1 0 1 1 1
  1 0 1 0 1
  1 0 1 0 1
  1 0 0 0 1
  1 1 1 1 1 (End)
It appears that a(n+1) is the maximum number of consecutive integers (beginning with 1) that can be placed, one after another, on an n-peg Towers of Hanoi, such that the sum of any two consecutive integers on any peg is a square. See the problem: http://online-judge.uva.es/p/v102/10276.html. - Ashutosh Mehra, Dec 06 2008
a(n) = number of (w,x,y) with all terms in {0,...,n} and w = |x+y-w|. - Clark Kimberling, Jun 11 2012
The same sequence also represents the solution to the "pigeons problem": maximal value of the sum of the lengths of n-1 line segments (connected at their end-points) required to pass through n trail dots, with unit distance between adjacent points, visiting all of them without overlaping two or more segments. In this case, a(0)=0, a(1)=1, a(2)=3, and so on. - Marco Ripà, Jan 28 2014
Also the longest path length in the n X n white bishop graph. - Eric W. Weisstein, Mar 27 2018
a(n) is the number of right triangles with sides n*(h-floor(h)), floor(h) and h, where h is the hypotenuse. - Andrzej Kukla, Apr 14 2021

			x^2 + 3*x^3 + 7*x^4 + 11*x^5 + 17*x^6 + 23*x^7 + 31*x^8 + 39*x^9 + 49*x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, Disorders and permutations , 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.
Graham Cormode, Notes on the organization number of a permutation.
Eric Weisstein's World of Mathematics, Longest Path Problem.
Eric Weisstein's World of Mathematics, White Bishop Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Complement of A047839. First difference is A052928.
Cf. A000217, A007590, A080827, A134151, A135151, A135152, A188653.
Partial sums: A213759(n-1) for n > 1. - Guenther Schrack, May 12 2018

[Floor(n^2/2)-1 : n in [2..100]]; // Wesley Ivan Hurt, Aug 06 2015

seq(floor((n^2+4*n+2)/2), n=0..20) # Gary Detlefs, Feb 10 2010

Table[Floor[n^2/2] - 1, {n, 2, 60}] (* Robert G. Wilson v, Aug 31 2006 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 3, 7, 11}, 60] (* Harvey P. Dale, Jan 16 2015 *)
Floor[Range[2, 20]^2/2] - 1 (* Eric W. Weisstein, Mar 27 2018 *)
Table[((-1)^n + 2 n^2 - 5)/4, {n, 2, 20}] (* Eric W. Weisstein, Mar 27 2018 *)
CoefficientList[Series[(-1 - x - x^2 + x^3)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *)

PARI
```
a(n) = n^2\2 - 1
```

a(2)=1; for n > 2, a(n) = a(n-1) + n - 1 + (n-1 mod 2). - Benoit Cloitre, Jan 12 2003
a(n) = T(n-1) + floor(n/2) - 1 = T(n) - floor((n+3)/2), where T(n) is the n-th triangular number (A000217). - Robert G. Wilson v, Aug 31 2006
Equals (n-1)-th row sums of triangles A134151 and A135152. Also, = binomial transform of [1, 2, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Nov 21 2007
G.f.: x^2*(1+x+x^2-x^3)/((1-x)^3*(1+x)). - R. J. Mathar, Sep 09 2008
a(n) = floor((n^2 + 4*n + 2)/2). - Gary Detlefs, Feb 10 2010
a(n) = abs(A188653(n)). - Reinhard Zumkeller, Apr 13 2011
a(n) = (2*n^2 + (-1)^n - 5)/4. - Bruno Berselli, Sep 14 2011
a(n) = a(-n) = A007590(n) - 1.
a(n) = A080827(n) - 2. - Kevin Ryde, Aug 24 2013
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4. - Wesley Ivan Hurt, Aug 06 2015
a(n) = A000217(n-1) + A004526(n-2), for n > 1. - J. Stauduhar, Oct 20 2017
From Guenther Schrack, May 12 2018: (Start)
Set a(0) = a(1) = -1, a(n) = a(n-2) + 2*n - 2 for n > 1.
a(n) = A000982(n-1) + n - 2 for n > 1.
a(n) = 2*A033683(n) - 3 for n > 1.
a(n) = A061925(n-1) + n - 3 for n > 1.
a(n) = A074148(n) - n - 1 for n > 1.
a(n) = A105343(n-1) + n - 4 for n > 1.
a(n) = A116940(n-1) - n for n > 1.
a(n) = A179207(n) - n + 1 for n > 1.
a(n) = A183575(n-2) + 1 for n > 2.
a(n) = A265284(n-1) - 2*n + 1 for n > 1.
a(n) = 2*A290743(n) - 5 for n > 1. (End)
E.g.f.: 1 + x + ((x^2 + x - 2)*cosh(x) + (x^2 + x - 3)*sinh(x))/2. - Stefano Spezia, May 06 2021
Sum_{n>=2} 1/a(n) = 3/2 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)) - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - Amiram Eldar, Sep 15 2022

Edited by Charles R Greathouse IV, Apr 23 2010

A028723 a(n) = (1/4)floor(n/2)floor((n-1)/2)floor((n-2)/2)floor((n-3)/2).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315, 441, 588, 784, 1008, 1296, 1620, 2025, 2475, 3025, 3630, 4356, 5148, 6084, 7098, 8281, 9555, 11025, 12600, 14400, 16320, 18496, 20808, 23409, 26163, 29241, 32490, 36100, 39900, 44100, 48510, 53361, 58443

Offset: 0

N. J. A. Sloane

It is not known whether A000241 and this sequence agree.
Conjectured to be crossing number of complete graph K_n, see A000241.
a(n+1) is the maximum number of rectangles that can be formed from n lines. - Erich Friedman
Number of symmetric Dyck paths of semilength n and having five peaks. E.g., a(6)=3 because we have U*DU*DUU*DDU*DU*D, U*DUU*DU*DU*DDU*D and UU*DU*DU*DU*DU*DD, where U=(1,1), D=(1,-1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004
a(n-5) is the number of length n words, w(1), w(2), ..., w(n) on alphabet {0,1,2} such that w(i) >= w(i+2) for all i. - Geoffrey Critzer, Mar 15 2014
a(n-1) is the number of length n binary strings beginning with a 1 that have exactly two pairs of consecutive 0's and two pairs of consecutive 1's. - Jeremy Dover, Jul 04 2016
Consider the partitions of n into two parts (p,q). Then 2*a(n+2) represents the total volume of all rectangular prisms with dimensions p, q and |q - p|. - Wesley Ivan Hurt, Apr 12 2018
a(n+1) is the number of subsets of {1, 2, ..., n} that contain 2 odd and 2 even numbers.  For example, for n = 6, a(7) = 9 and the 9 subsets are {1,2,3,4}, {1,2,3,6}, {1,2,4,5}, {1,2,5,6}, {1,3,4,6}, {1,4,5,6}, {2,3,4,5}, {2,3,5,6}, {3,4,5,6}. - Enrique Navarrete, Dec 22 2019
a(n+1) is the maximum number of induced 4-cycles in an n-node graph (Pippenger and Golumbic 1975). - Pontus von Brömssen, Mar 27 2022

			G.f. = x^5 + 3*x^6 + 9*x^7 + 18*x^8 + 36*x^9 + 60*x^10 + 100*x^11 + ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.18, p. 533.
Martin Gardner, Knotted Doughnuts and Other Mathematical Entertainments, W. H. Freeman & Company, 1986, Chapter 11, pages 133-144.
Carsten Thomassen, Embeddings and Minors, in: R. L. Graham, M. Grötschel, and L. Lovász, Handbook of Combinatorics, Vol. 1, Elsevier, 1995, p. 314.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, and Gelasio Salazar, The 2-Page Crossing Number of K_n, arXiv:1206.5669 [math.CO], 2012.
Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, and Gelasio Salazar, The 2-Page Crossing Number of K_n, Discrete Comput. Geom., Vol. 49, No. 4 (2013), pp. 747-777. MR3068573.
James Dolan et al., su(3) and Zarankiewicz's conjecture.
Dhruv Mubayi, Counting substructures II: Hypergraphs, Combinatorica, Vol. 33, No. 5 (2013), pp. 591--612. MR3132928
Nicholas Pippenger and Martin Charles Golumbic, The inducibility of graphs, Journal of Combinatorial Theory Series B 19 (1975), 189-203.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A000241, A006918.

[(n^4-8*n^3+18*n^2-12*n+2*n*(n-2)*((1+(-1)^n)/2)+(2*n-3)^2*((1-(-1)^n)/2))/64: n in [0..50]]; // Vincenzo Librandi, Mar 23 2014

A028723:=n->(1/4)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2); seq(A028723(n), n=0..100); # Wesley Ivan Hurt, Nov 01 2013

Table[If[EvenQ[n], n(n-2)^2(n-4)/64, (n-1)^2(n-3)^2/64], {n, 0, 50}]
Table[(n^4 -8n^3 +18n^2 -12n + 2n(n-2)((1+(-1)^n)/2) +(2n-3)^2((1-(-1)^n)/2))/64, {n, 0, 50}] (* Vincenzo Librandi, Mar 23 2014 *)
LinearRecurrence[{2, 2,-6,0,6,-2,-2,1}, {0,0,0,0,0,1,3,9}, 50] (* Harvey P. Dale, Sep 13 2018 *)
Times@@@Table[Floor[(n-k)/2], {n,0,60}, {k,0,3}]/4 (* Eric W. Weisstein, Apr 29 2019 *)

a(n) = if (n % 2, (n-1)^2 *(n-3)^2/64, n*(n-2)^2 *(n-4)/64); \\ Michel Marcus, Nov 02 2013

{a(n) = prod(k=0, 3, (n - k) \ 2) / 4}; /* Michael Somos, Nov 02 2014 */

[(n*(-12 +18*n -8*n^2 +n^3) +2*n*(n-2)*((n+1)%2) +(2*n-3)^2*(n%2))/64 for n in (0..60)] # G. C. Greubel, Apr 08 2022

If n even, n*(n-2)^2*(n-4)/64; if n odd, (n-1)^2*(n-3)^2/64.
G.f.: x^5*(1+x+x^2)/((1-x)^5*(1+x)^3). - Emeric Deutsch, Jan 12 2004
For n>2, a(n) = A007590(n-3)*A007590(n-1)/16. - Richard R. Forberg, Dec 03 2013
a(n) = (n^4 -8*n^3 +18*n^2 -12*n +2*n*(n-2)*((1+(-1)^n)/2) + (2*n-3)^2*((1-(-1)^n)/2))/64. - Luce ETIENNE, Mar 22 2014
Euler transform of length 3 sequence [3, 3, -1]. - Michael Somos, Nov 02 2014
a(n) = a(4-n) for all n in Z. - Michael Somos, Nov 02 2014
0 = -3 + a(n) - a(n+1) - 3*a(n+2) + 3*a(n+3) + 3*a(n+4) - 3*a(n+5) - a(n+6) + a(n+7) for all n in Z. - Michael Somos, Nov 02 2014
0 = a(n)*(+a(n+2) + a(n+3)) + a(n+1)*(-3*a(n+2) +a(n+3)) for all n in Z. - Michael Somos, Nov 02 2014
a(n+1)^2 - a(n)*a(n+2) = binomial(n/2, 2)^3 for all even n in Z ( = 0 if n odd). - Michael Somos, Nov 02 2014
a(n)*(a(n+1) + a(n+2)) +a(n+1)*(-3*a(n+1) + a(n+2)) = 0 for all even n in Z ( = k^4 * (k^2 - 1) / 4 if n = 2*k + 1). - Michael Somos, Nov 02 2014
a(n) = binomial(n/2,2)^2, n even; a(n) = binomial((n-1)/2,2)*binomial((n+1)/2,2), n odd. - Enrique Navarrete, Dec 22 2019
E.g.f.: (1/128)*exp(-x)*(exp(2*x)*(9 - 12*x + 8*x^2 - 4*x^3 + 2*x^4) - 9 - 6*x - 2*x^2). - Stefano Spezia, Dec 27 2019
a(n) = A002620(n-1)*A002620(n-3)/4. - R. J. Mathar, Mar 23 2021
a(n)= A096338(n-6)+A096338(n-5)+A096338(n-4). - R. J. Mathar, Mar 23 2021
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=5} 1/a(n) = 2*Pi^2/3 - 5.
Sum_{n>=5} (-1)^(n+1)/a(n) = 2*Pi^2 - 19. (End)

A056827 a(n) = floor(n^2/6).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 6, 8, 10, 13, 16, 20, 24, 28, 32, 37, 42, 48, 54, 60, 66, 73, 80, 88, 96, 104, 112, 121, 130, 140, 150, 160, 170, 181, 192, 204, 216, 228, 240, 253, 266, 280, 294, 308, 322, 337, 352, 368, 384, 400, 416, 433, 450, 468, 486, 504

Offset: 0

N. J. A. Sloane, Sep 02 2000

a(n-1) represents the floor of the area under the polygon connecting the lattice points (n, floor(n/3)) from 0..n, n>0 (see example). - Wesley Ivan Hurt, Jun 06 2014

			5|                                             .__.__.
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4|                                    .__.__./_|__|__|
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3|                           .__.__./_|__|__|__|__|__|
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2|                  .__.__./_|__|__|__|__|__|__|__|__|
.|                 /|  |  |  |  |  |  |  |  |  |  |  |
1|         .__.__./_|__|__|__|__|__|__|__|__|__|__|__|
.|        /|  |  |  |  |  |  |  |  |  |  |  |  |  |  |
0|.__.__./_|__|__|__|__|__|__|__|__|__|__|__|__|__|__|_________________
  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17   .. n
  0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42 48   .. a(n)
     0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42   .. a(n-1) <--

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. A000290, A007590, A000212, A002620, A118015, A118013, A056834, A130519, A056838, A056865.

List([0..60], n-> Int(n^2/6) ); # G. C. Greubel, Jul 23 2019

[Floor(n^2/6): n in [0..60]]; // Vincenzo Librandi, May 08 2011

A056827:=n->floor(n^2/6); seq(A056827(k), k=0..60); # Wesley Ivan Hurt, Oct 29 2013

Floor[Range[0,60]^2/6] (* or *) LinearRecurrence[{2,-1,0,0,0,1,-2,1}, {0,0,0,1,2,4,6,8}, 60] (* Harvey P. Dale, Jun 06 2013 *)

n^2\6 \\ Charles R Greathouse IV, May 08 2011

[floor(n^2/6) for n in (0..60)] # G. C. Greubel, Jul 23 2019

From R. J. Mathar, Nov 22 2008: (Start)
G.f.: x^3*(1+x^2)/((1+x)*(1-x)^3*(1+x+x^2)*(1-x+x^2)).
a(n+1) - a(n) = A123919(n). (End)
a(n) = floor( (1/2) * Sum_{i=1..n+1} (ceiling(i/3) + floor(i/3) - 1) ). - Wesley Ivan Hurt, Jun 06 2014
Sum_{n>=3} 1/a(n) = 15/8 + Pi^2/36 - Pi/(4*sqrt(3)) + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Aug 13 2022

A118015 a(n) = floor(n^2/5).

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 7, 9, 12, 16, 20, 24, 28, 33, 39, 45, 51, 57, 64, 72, 80, 88, 96, 105, 115, 125, 135, 145, 156, 168, 180, 192, 204, 217, 231, 245, 259, 273, 288, 304, 320, 336, 352, 369, 387, 405, 423, 441, 460, 480, 500, 520, 540, 561, 583, 605, 627, 649, 672

Offset: 0

Reinhard Zumkeller, Apr 10 2006

It seems that for n >= 5, a(n) is the maximum number of non-overlapping 1 X 5 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's program. - Dmitry Kamenetsky, Aug 03 2009

Ismailescu & Lee prove that for n > 6, a(n) is composite. - Charles R Greathouse IV, Jan 10 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Dan Ismailescu and Yunkyu James Lee, Polynomially growing integer sequences all whose terms are composite, arXiv:2501.04851 [math.NT], 2025. See p. 1.
R. D. Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A000212, A000290, A007590, A002620, A056827, A118013, A279169.

Cf. A056834, A130519, A056838, A056865.

[ n^2 div 5: n in [0..58] ]; // Klaus Brockhaus, Nov 18 2008

Table[Floor[n^2/5], {n, 0, 60}] (* Bruno Berselli, Dec 12 2016 *)

a(n)=n^2\5 \\ Charles R Greathouse IV, Sep 24 2015

[int(n**2/5) for n in range(60)] # Bruno Berselli, Dec 12 2016

[floor(n^2/5) for n in range(60)] # Bruno Berselli, Dec 12 2016

G.f.: x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4)*(1 - x)^3). - Klaus Brockhaus, Nov 18 2008
a(n) = A008732(n-4) + A008732(n-3). - R. J. Mathar, Nov 22 2008
a(5*m+r) = m*(5*m + 2*r) + a(r), with m >= 0 and 0 <= r < 5. Example: for m=4 and r=3, a(5*4+3) = a(23) = 4*(5*4 + 2*3) + a(3) = 104 + 1 = 105. - Bruno Berselli, Dec 12 2016
Sum_{n>=3} 1/a(n) = 25/16 + Pi^2/30 + sqrt(5-2*sqrt(5))*Pi/4. - Amiram Eldar, Aug 13 2022

Edited by Charles R Greathouse IV, Apr 20 2010

A156040 Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.

Original entry on oeis.org

1, 1, 3, 4, 6, 8, 11, 13, 17, 20, 24, 28, 33, 37, 43, 48, 54, 60, 67, 73, 81, 88, 96, 104, 113, 121, 131, 140, 150, 160, 171, 181, 193, 204, 216, 228, 241, 253, 267, 280, 294, 308, 323, 337, 353, 368, 384, 400, 417, 433, 451, 468, 486, 504, 523, 541, 561, 580, 600

Offset: 0

Jack W Grahl, Feb 02 2009, Feb 11 2009

For n = 1, 2 these are just the triangular numbers. a(n) is always at least 1/3 of the corresponding triangular number, since each partition of this type gives up to three ordered partitions with the same cyclical order.
An alternative definition, which avoids using parts of size 0: a(n) is the third diagonal of A184957. - N. J. A. Sloane, Feb 27 2011
Diagonal sums of the triangle formed by rows T(2, k) k = 0, 1, ..., 2m of ascending m-nomial triangles (see A004737):
1
1 2 1
1 2 3 2 1
1 2 3 4 3 2 1
1 2 3 4 5 4 3 2 1
1 2 3 4 5 6 5 4 3 2 1
- Bob Selcoe, Feb 07 2014
Arrange A004396 in rows successively shifted to the right two spaces and sum the columns:
1  1  2  3  3  4  5  5  6 ...
      1  1  2  3  3  4  5 ...
            1  1  2  3  3 ...
                  1  1  2 ...
                        1 ...
------------------------------
1  1  3  4  6  8 11 13 17 ... - L. Edson Jeffery, Jul 30 2014
a(n) is the dimension of three-dimensional (2n + 2)-homogeneous polynomial vector fields with full tetrahedral symmetry (for a given orthogonal representation), and which are solenoidal. - Giedrius Alkauskas, Sep 30 2017
Also the number of compositions of n + 3 into three parts, the first at least as great as each of the other two. Also the number of compositions of n + 4 into three parts, the first strictly greater than each of the other two. - Gus Wiseman, Oct 09 2020

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + 20*x^9 + ...
The a(4) = 6 compositions of 4 are: (4 0 0), (3 1 0), (3 0 1), (2 2 0), (2 1 1), (2 0 2).
From _Gus Wiseman_, Oct 05 2020: (Start)
The a(0) = 1 through a(7) = 13 triples of nonnegative integers summing to n where the first is at least as great as each of the other two are:
  (000)  (100)  (101)  (111)  (202)  (212)  (222)  (313)
                (110)  (201)  (211)  (221)  (303)  (322)
                (200)  (210)  (220)  (302)  (312)  (331)
                       (300)  (301)  (311)  (321)  (403)
                              (310)  (320)  (330)  (412)
                              (400)  (401)  (402)  (421)
                                     (410)  (411)  (430)
                                     (500)  (420)  (502)
                                            (501)  (511)
                                            (510)  (520)
                                            (600)  (601)
                                                   (610)
                                                   (700)
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Giedrius Alkauskas, Projective and polynomial superflows. I, arxiv.org/1601.06570 [math.AG] (2017), Section 4.3.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

For compositions into 4 summands see A156039; also see A156041 and A156042.
Cf. A184957, A071619 (bisection).
A001399(n-2)*2 is the strict case.
A001840(n-2) is the version with opposite relations.
A001840(n-1) is the version with strict opposite relations.
A069905 is the case with strict relations.
A014311 ranks 3-part compositions, with strict case A337453.
A014612 ranks 3-part partitions, with strict case A007304.
Cf. A000217, A001523, A211540, A218004, A220377, A337483, A337484.

a:= proc(n) local m, r; m := iquo(n, 6, 'r'); (4 +6*m +2*r) *m + [1, 1, 3, 4, 6, 8][r+1] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009

nn = 58; CoefficientList[Series[x^3/(1 - x^2)^2/(1 - x^3) + 1/(1 - x^2)^2/(1 - x), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 14 2013 *)
CoefficientList[Series[(1 + x^2)/((1 + x) * (1 + x + x^2) * (1 - x)^3), {x, 0, 58}], x] (* L. Edson Jeffery, Jul 29 2014 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 1, 3, 4, 6, 8}, 60] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+3,{3}],#[[1]]>=#[[2]]&&#[[1]]>=#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020*)

{a(n) = n*(n+4)\6 + 1}; /* Michael Somos, Mar 26 2017 */

G.f.: (x^2+1) / (1-x-x^2+x^4+x^5-x^6). - Alois P. Heinz, Jun 14 2009
Slightly nicer g.f.: (1+x^2)/((1-x)*(1-x^2)*(1-x^3)). - N. J. A. Sloane, Apr 29 2011
a(n) = A007590(n+2) - A000212(n+2). - Richard R. Forberg, Dec 08 2013
a(2*n) = A071619(n+1). - L. Edson Jeffery, Jul 29 2014
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 6, a(5) = 8. - Harvey P. Dale, May 28 2015
a(n) = (n^2 + 4*n + 3)/6 + IF(MOD(n, 2) = 0, 1/2) + IF(MOD(n, 3) = 1, -1/3). - Heinrich Ludwig, Mar 21 2017
a(n) = 1 + floor((n^2 + 4*n)/6). - Giovanni Resta, Mar 21 2017
Euler transform of length 4 sequence [1, 2, 1, -1]. - Michael Somos, Mar 26 2017
a(n) = a(-4 - n) for all n in Z. - Michael Somos, Mar 26 2017
0 = a(n)*(-1 + a(n) - 2*a(n+1) - 2*a(n+2) + 2*a(n+3)) + a(n+1)*(+1 + a(n+1) + 2*a(n+2) - 2*a(n+3)) + a(n+2)*(+1 + a(n+2) - 2*a(n+3)) + a(n+3)*(-1 + a(n+3)) for all n in Z. - Michael Somos, Mar 26 2017
a(n) = round((n+1)*(n+3)/6). - Bill McEachen, Feb 16 2021
Sum_{n>=0} 1/a(n) = 3/2 + Pi^2/36 + (tan(c1)-1)*c1 + 3*c2*sinh(c2)/(1+2*cosh(c2)), where c1 = Pi/(2*sqrt(3)) and c2 = Pi*sqrt(2)/3. - Amiram Eldar, Dec 10 2022
E.g.f.: ((16 + 15*x + 3*x^2)*cosh(x) + 2*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (7 + 15*x + 3*x^2)*sinh(x))/18. - Stefano Spezia, Apr 05 2023

More terms from Alois P. Heinz, Jun 14 2009

A001972 Expansion of 1/((1-x)^2(1-x^4)) = 1/( (1+x)(1+x^2)*(1-x)^3 ).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 128, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 242, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 392, 406, 420, 435, 450, 465

Offset: 0

N. J. A. Sloane

First differences are A008621. - Amarnath Murthy, Apr 26 2004
a(n) = least k > a(n-1) such that k + a(n-1) + a(n-2) + a(n-3) is triangular. - Amarnath Murthy, Apr 26 2004
From Jon Perry, Nov 16 2010: (Start)
Column sums of the following array:
  1 2 3 4 5 6 7  8  9...
          1 2 3  4  5...
                    1...
  --------------------
  1 2 3 4 6 8 10 12 15 (End)
A001972(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2=4x+y. - Clark Kimberling, Jun 04 2012
Number of partitions of n into parts 1 (of two sorts) and 4 (of one sort). - Joerg Arndt, Aug 08 2013
In the polynomial sequence s(n) = (x*s(n-1)*s(n-4) + y*s(n-2)*s(n-3))/s(n-5), with s(k) = 1 for k = 0..4, the leading term of s(n+5) is x^a(n). See A333260. - Michael Somos, Mar 13 2020

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 208
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=4]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Bisections are A000217 and A007590. - Amarnath Murthy, Apr 26 2004

Cf. A001972, A002620, A008621, A056594, A130519, A333260.

[Floor((n+3)^2/8): n in [0..60]]; // Vincenzo Librandi, Aug 15 2011

A001972:=-(2-z+z**3-2*z**4+z**5)/(z+1)/(z**2+1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 1

CoefficientList[Series[1/((1-x)^2(1-x^4)),{x,0,80}],x]  (* Harvey P. Dale, Mar 27 2011 *)

PARI
```
a(n)=(n+3)^2\8;
```

From Michael Somos, Apr 21 2000: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) + 1.
a(n) = floor((n+3)^2/8). (End)
a(n) = Sum_{k=0..n} floor((k+4)/4) = n + 1 + Sum_{k=0..n} floor(k/4). - Paul Barry, Aug 19 2003
a(n) = a(n-4) + n + 1. - Paul Barry, Jul 14 2004
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+4} floor(j/4);
a(n-4) = (1/2)*floor(n/4)*(2*n - 2 - 4*floor(n/4)). (End)
A002620(n+1) = a(2*n-1)/2.
A000217(n+1) = a(2*n).
a(n)+a(n+1)+a(n+2)+a(n+3) = (n+4)*(n+5)/2. - Amarnath Murthy, Apr 26 2004
a(n) = n^2/8 + 3*n/4 + 15/16 + (-1)^n/16 + A056594(n+3)/4. - Amarnath Murthy, Apr 26 2004
a(n) = A130519(n+4). - Franklin T. Adams-Watters, Jul 10 2009
a(n) = floor((n+1)/(1-e^(-8/(n+1)))). - Richard R. Forberg, Aug 07 2013
a(n) = a(-6-n) for all n in Z. - Michael Somos, Mar 13 2020
E.g.f.: ((8 + 7*x + x^2)*cosh(x) + 2*sin(x) + (7 + 7*x + x^2)*sinh(x))/8. - Stefano Spezia, May 09 2023

Partially edited by R. J. Mathar, Jul 11 2009

A061925 a(n) = ceiling(n^2/2) + 1.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926, 969, 1014, 1059, 1106, 1153, 1202, 1251, 1302, 1353, 1406

Offset: 0

Henry Bottomley, May 17 2001

a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004
First term in each group in A074148. - Amarnath Murthy, Aug 28 2002
From Christian Barrientos, Jan 01 2021: (Start)
For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.
For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:
      _       _       _       _       _      
   |||_|   |||_|   |||_|   |||_|   |||_|   |||_
   |||_|   |||     || ||       ||     ||       |||
(End)
Except for a(2), a(n) agrees with the lower matching number of the (n+1) X (n+1) bishop graph up to at least n = 13. - Eric W. Weisstein, Dec 23 2024

Harry J. Smith, Table of n, a(n) for n = 0..1000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Eric Weisstein's World of Mathematics, Bishop Graph.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A100795, A074147-A074149.

seq(floor((n^2+3)/2),n=0..25); # Gary Detlefs, Feb 13 2010

Table[Ceiling[n^2/2]+1,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
LinearRecurrence[{2,0,-2,1},{1,2,3,6},60] (* Harvey P. Dale, Jan 03 2024 *)

a(n) = { ceil(n^2/2) + 1 } \\ Harry J. Smith, Jul 29 2009

a(n) = a(n-1) + 2*floor((n-1)/2) + 1 = A061926(3, k) = 2*A002620(n+1) - (n-1) = A000982(n) + 1.
a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).
a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).
a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010
From R. J. Mathar, Feb 19 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)
a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011
a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013
a(n) + a(n+1) = A027688(n). a(n+1) - a(n) = A109613(n). - R. J. Mathar, Jul 20 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007

A033436 a(n) = ceiling( (3*n^2 - 4)/8 ).

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 13, 18, 24, 30, 37, 45, 54, 63, 73, 84, 96, 108, 121, 135, 150, 165, 181, 198, 216, 234, 253, 273, 294, 315, 337, 360, 384, 408, 433, 459, 486, 513, 541, 570, 600, 630, 661, 693, 726, 759, 793, 828

Offset: 0

N. J. A. Sloane

Number of edges in 4-partite Turan graph of order n.

Apart from the initial term this equals the elliptic troublemaker sequence R_n(1,4) (also sequence R_n(3,4)) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 08 2013

R. L. Graham, Martin Grötschel, and László Lovász, Handbook of Combinatorics, Vol. 2, 1995, p. 1234.

Ivan Panchenko, Table of n, a(n) for n = 0..10000
Kevin Beanland, Hung Viet Chu, and Carrie E. Finch-Smith, Generalized Schreier sets, linear recurrence relation, Turán graphs, arXiv:2112.14905 [math.CO], 2021.
Katherine E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials, arXiv:1108.3051 [math.NT], 2011-2014.
Eric Weisstein's World of Mathematics, Turán Graph.
Wikipedia, Turán graph.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A002620 (= R_n(1,2)), A000212 (= R_n(1,3) = R_n(2,3)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A033440, A033441, A033442, A033443, A033444.

Cf. A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A184535 (= R_n(2,5) = R_n(3,5)).

LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 0, 1, 3, 6, 9}, 48] (* Jean-François Alcover, Sep 21 2017 *)

a(n)=(3*n^2 +3)\8 \\ Charles R Greathouse IV, Sep 24 2015

The second differences of the listed terms are periodic with period (1, 1, 1, 0) of length 4, showing that the terms satisfy the recurrence a(n) = 2a(n-1)-a(n-2)+a(n-4)-2a(n-5)+a(n-6). - John W. Layman, Jan 23 2001
a(n) = (1/16) {6n^2 - 5 + (-1)^n + 2(-1)^[n/2] - 2(-1)^[(n-1)/2] }. Therefore a(n) is asymptotic to 3/8*n^2. - Ralf Stephan, Jun 09 2005
O.g.f.: -x^2*(1+x+x^2)/((x+1)*(x^2+1)*(x-1)^3). - R. J. Mathar, Dec 05 2007
a(n) = Sum_{k=0..n} A166486(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
a(n) = floor(3*n^2/8). - Peter Bala, Aug 08 2013
a(n) = Sum_{i=1..n} floor(3*i/4). - Wesley Ivan Hurt, Sep 12 2017
Sum_{n>=2} 1/a(n) = Pi^2/36 + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)) + 2/3. - Amiram Eldar, Sep 24 2022

A030511 Graham-Sloane-type lower bound on the size of a ternary (n,3,3) constant-weight code.

Original entry on oeis.org

2, 6, 10, 16, 24, 32, 42, 54, 66, 80, 96, 112, 130, 150, 170, 192, 216, 240, 266, 294, 322, 352, 384, 416, 450, 486, 522, 560, 600, 640, 682, 726, 770, 816, 864, 912, 962, 1014, 1066, 1120, 1176, 1232, 1290, 1350, 1410, 1472, 1536, 1600

Offset: 3

Mattias Svanstrom (mattias(AT)isy.liu.se)

With a different offset this is the elliptic troublemaker sequence R_n(2,6) (also sequence R_n(4,6)) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 12 2013

a(n) is the maximum number of equilateral triangles that can be formed by adding n+1 straight lines on an infinite grid of regular hexagons. - Dhairya Baxi, Sep 03 2022

Craig Knecht, Maximum number of enneiamonds in a hexagon.
Katherine E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051 [math.NT], 2011-2014.
M. Svanstrom, A lower bound for ternary constant weight codes, IEEE Trans. on Information Theory, Vol. 43, No. 5 (Sep. 1997), pp. 1630-1632.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A002620 (= R_n(1,2)), A007590 (= R_n(2,4)), A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A184535 (= R_n(2,5) = R_n(3,5)).

LinearRecurrence[{2,-1,1,-2,1},{2,6,10,16,24},50] (* Harvey P. Dale, Mar 03 2016 *)

def A030511(n): return ((n-1)**2<<1)//3 # Chai Wah Wu, Aug 04 2025

a(n) = 2 * (n - 1)^2 / 3 if n==1 (mod 3), a(n) = 2 * n * (n - 2) / 3 otherwise.
G.f.: -2*x^3*(1 + x) / ( (1 + x + x^2)*(x - 1)^3 ). - R. J. Mathar, Aug 25 2011
a(n) = 2*A000212(n-1). - R. J. Mathar, Aug 25 2011
a(n) = floor( (2/3)*(n-1)^2 ). - Wesley Ivan Hurt, Jun 19 2013
a(n) = (2*(n - 2)*n - (-1)^floor(2*(n-2)/3) + 1)/3. - Bruno Berselli, Aug 08 2013
a(n) = a(n-1) + 2*floor((n-1)*2/3). - Gionata Neri, Apr 26 2015
a(n) = floor((n-2)*(n-1)/3) + floor((n-1)*n/3) = floor((n-1)*(n+1)/3) + floor((n-1)*(n-3)/3). - Bruno Berselli, Mar 02 2017
Sum_{n>=3} 1/a(n) = Pi^2/36 + Pi/(4*sqrt(3)) + 3/8. - Amiram Eldar, Sep 24 2022
E.g.f.: 2*exp(-x/2)*(exp(3*x/2)*(1 + 3*x*(x - 1)) - cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 28 2022

A033437 Number of edges in 5-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 14, 19, 25, 32, 40, 48, 57, 67, 78, 90, 102, 115, 129, 144, 160, 176, 193, 211, 230, 250, 270, 291, 313, 336, 360, 384, 409, 435, 462, 490, 518, 547, 577, 608, 640, 672, 705, 739, 774, 810, 846, 883, 921, 960, 1000, 1040, 1081, 1123, 1166, 1210, 1254

Offset: 0

N. J. A. Sloane

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,5) (also sequence R_n(4,5)) in the notation of Stange (see Table 1, p. 16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 12 2013

R. L. Graham et al., eds., Handbook of Combinatorics, Vol. 2, p. 1234.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Kevin Beanland, Hung Viet Chu, and Carrie E. Finch-Smith, Generalized Schreier sets, linear recurrence relation, Turán graphs, arXiv:2112.14905 [math.CO], 2021.
K. E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051 [math.NT], 2011-2014.
Eric Weisstein's World of Mathematics, Turán Graph
Wikipedia, Turán graph
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033438, A033439, A033440, A033441, A033442, A033443, A033444. - Reinhard Zumkeller, Nov 30 2009
Elliptic troublemaker sequences: A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A184535 (= R_n(2,5) = R_n(3,5)).
Cf. A279169.

[2*n^2 div 5: n in [0..60]]; // Vincenzo Librandi, Apr 20 2015

Mathematica
```
Table[Floor[2n^2/5],{n,0,60}]
```

a(n)=2*n^2\5 \\ Charles R Greathouse IV, Apr 20 2015

G.f.: (x^5+x^4+x^3+x^2)/((1-x^5)*(1-x)^2).
a(n) = Sum_{k=0..n} A011558(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
a(n) = floor( 2n^2/5 ). - Wesley Ivan Hurt, Jun 20 2013
a(n) = Sum_{i=1..n} floor(4*i/5). - Wesley Ivan Hurt, Sep 12 2017

cp's OEIS Frontend

A047838 a(n) = floor(n^2/2) - 1.

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A028723 a(n) = (1/4)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2).

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A056827 a(n) = floor(n^2/6).

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A118015 a(n) = floor(n^2/5).

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A156040 Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.

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A001972 Expansion of 1/((1-x)^2*(1-x^4)) = 1/( (1+x)*(1+x^2)*(1-x)^3 ).

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A028723 a(n) = (1/4)floor(n/2)floor((n-1)/2)floor((n-2)/2)floor((n-3)/2).

A001972 Expansion of 1/((1-x)^2(1-x^4)) = 1/( (1+x)(1+x^2)*(1-x)^3 ).