A033442 -id:A033442 - cp's OEIS frontend

A000212 a(n) = floor(n^2/3).

0, 0, 1, 3, 5, 8, 12, 16, 21, 27, 33, 40, 48, 56, 65, 75, 85, 96, 108, 120, 133, 147, 161, 176, 192, 208, 225, 243, 261, 280, 300, 320, 341, 363, 385, 408, 432, 456, 481, 507, 533, 560, 588, 616, 645, 675, 705, 736, 768, 800, 833, 867, 901, 936

Offset: 0

N. J. A. Sloane

Let M_n be the n X n matrix of the following form: [3 2 1 0 0 0 0 0 0 0 / 2 3 2 1 0 0 0 0 0 0 / 1 2 3 2 1 0 0 0 0 0 / 0 1 2 3 2 1 0 0 0 0 / 0 0 1 2 3 2 1 0 0 0 / 0 0 0 1 2 3 2 1 0 0 / 0 0 0 0 1 2 3 2 1 0 / 0 0 0 0 0 1 2 3 2 1 / 0 0 0 0 0 0 1 2 3 2 / 0 0 0 0 0 0 0 1 2 3]. Then for n > 2 a(n) = det M_(n-2). - Benoit Cloitre, Jun 20 2002
Largest possible size for the directed Cayley graph on two generators having diameter n - 2. - Ralf Stephan, Apr 27 2003
It seems that for n >= 2, a(n) is the maximum number of non-overlapping 1 X 3 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 tool, see links. - Dmitry Kamenetsky, Aug 03 2009
Maximum number of edges in a K4-free graph with n vertices. - Yi Yang, May 23 2012
3a(n) + 1 = y^2 if n is not 0 mod 3 and 3a(n) = y^2 otherwise. - Jon Perry, Sep 10 2012
Apart from the initial term this is the elliptic troublemaker sequence R_n(1, 3) (also sequence R_n(2, 3)) 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
The number of partitions of 2n into exactly 3 parts. - Colin Barker, Mar 22 2015
a(n-1) is the maximum number of non-overlapping triples (i,k), (i+1, k+1), (i+2, k+2) in an n X n matrix. Details: The triples are distributed along the main diagonal and 2*(n-1) other diagonals. Their maximum number is floor(n/3) + 2*Sum_{k = 1..n-1} floor(k/3) = floor((n-1)^2/3). - Gerhard Kirchner, Feb 04 2017
Conjecture:  a(n) is the number of intersection points of n cevians that cut a triangle into the maximum number of pieces (see A007980). - Anton Zakharov, May 07 2017
From Gus Wiseman, Oct 05 2020: (Start)
Also the number of unimodal triples (meaning the middle part is not strictly less than both of the other two) of positive integers summing to n + 1. The a(2) = 1 through a(6) = 12 triples are:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,2,4)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,3,3)
                    (2,2,1)  (1,4,1)  (1,4,2)
                    (3,1,1)  (2,2,2)  (1,5,1)
                             (2,3,1)  (2,2,3)
                             (3,2,1)  (2,3,2)
                             (4,1,1)  (2,4,1)
                                      (3,2,2)
                                      (3,3,1)
                                      (4,2,1)
                                      (5,1,1)
(End)

			G.f. = x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 21*x^8 + 27*x^9 + 33*x^10 + ...
From _Gus Wiseman_, Oct 07 2020: (Start)
The a(2) = 1 through a(6) = 12 partitions of 2*n into exactly 3 parts (Barker) are the following. The Heinz numbers of these partitions are given by the intersection of A014612 (triples) and A300061 (even sum).
  (2,1,1)  (2,2,2)  (3,3,2)  (4,3,3)  (4,4,4)
           (3,2,1)  (4,2,2)  (4,4,2)  (5,4,3)
           (4,1,1)  (4,3,1)  (5,3,2)  (5,5,2)
                    (5,2,1)  (5,4,1)  (6,3,3)
                    (6,1,1)  (6,2,2)  (6,4,2)
                             (6,3,1)  (6,5,1)
                             (7,2,1)  (7,3,2)
                             (8,1,1)  (7,4,1)
                                      (8,2,2)
                                      (8,3,1)
                                      (9,2,1)
                                      (10,1,1)
(End)

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..5000
Kevin Beanland, Hung Viet Chu, and Carrie E. Finch-Smith, Generalized Schreier sets, linear recurrence relation, Turán graphs, arXiv:2112.14905 [math.CO], 2021.
Rafael Durbano Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem.
Bakir Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4.
Bakir Farhi, An Elementary Proof that any Natural Number can be Written as the Sum of Three Terms of the Sequence floor(n^2/3), Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.6.
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.
Katherine E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051v3 [math.NT], 2011-2014.
C. K. Wong and Don Coppersmith, A combinatorial problem related to multimodule memory organizations, J. ACM 21 (1974), 392-402.
Anton Zakharov, Cevians.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000290, A007590 (= R_n(2,4)), A002620 (= R_n(1,2)), A118015, A056827, A118013.
Cf. 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)), A033440, A033441, A033442, A033443, A033444.
Cf. A001353 and A004523 (first differences). A184535 (= R_n(2,5) = R_n(3,5)).
Cf. A238738. - Bruno Berselli, Apr 17 2015
Cf. A007980
Cf. A005408.
A000217(n-2) counts 3-part compositions.
A014612 ranks 3-part partitions, with strict case A007304.
A069905 counts the 3-part partitions.
A211540 counts strict 3-part partitions.
A337453 ranks strict 3-part compositions.
Cf. A056834, A130519, A056838, A056865.
A001399(n-6)*4 is the strict version.
A001523 counts unimodal compositions, with strict case A072706.
A001840(n-4) is the non-unimodal version.
A001399(n-6)*2 is the strict non-unimodal version.
A007052 counts unimodal patterns.
A115981 counts non-unimodal compositions, ranked by A335373.
A011782 counts unimodal permutations.
A335373 is the complement of a ranking sequence for unimodal compositions.
A337459 ranks these compositions, with complement A337460.
Cf. A069905, A220377, A332743, A337461, A337561, A337603, A337604.

[Floor(n^2 / 3): n in [0..50]]; // Vincenzo Librandi, May 08 2011

A000212:=(-1+z-2*z**2+z**3-2*z**4+z**5)/(z**2+z+1)/(z-1)**3; # Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence with an additional leading 1.
A000212 := proc(n) option remember; `if`(n<4, [0,0,1,3][n+1], a(n-1)+a(n-3) -a(n-4)+2) end; # Peter Luschny, Nov 20 2011

Table[Quotient[n^2, 3], {n, 0, 59}] (* Michael Somos, Jan 22 2014 *)

{a(n) = n^2 \ 3}; /* Michael Somos, Sep 25 2006 */

def A000212(n): return n**2//3 # Chai Wah Wu, Jun 07 2022

G.f.: x^2*(1+x)/((1-x)^2*(1-x^3)). - Franklin T. Adams-Watters, Apr 01 2002
Euler transform of length 3 sequence [ 3, -1, 1]. - Michael Somos, Sep 25 2006
G.f.: x^2 * (1 - x^2) / ((1 - x)^3 * (1 - x^3)). a(-n) = a(n). - Michael Somos, Sep 25 2006
a(n) = Sum_{k = 0..n} A011655(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
a(n) = a(n-1) + a(n-3) - a(n-4) + 2 for n >= 4. - Alexander Burstein, Nov 20 2011
a(n) = a(n-3) + A005408(n-2) for n >= 3. - Alexander Burstein, Feb 15 2013
a(n) = (n-1)^2 - a(n-1) - a(n-2) for n >= 2. - Richard R. Forberg, Jun 05 2013
Sum_{n >= 2} 1/a(n) = (27 + 6*sqrt(3)*Pi + 2*Pi^2)/36. - Enrique Pérez Herrero, Jun 29 2013
0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jan 22 2014
a(n) = Sum_{k = 1..n} k^2*A049347(n+2-k). - Mircea Merca, Feb 04 2014
a(n) = Sum_{i = 1..n+1} (ceiling(i/3) + floor(i/3) - 1). - Wesley Ivan Hurt, Jun 06 2014
a(n) = Sum_{j = 1..n} Sum_{i=1..n} ceiling((i+j-n-1)/3). - Wesley Ivan Hurt, Mar 12 2015
a(n) = Sum_{i = 1..n} floor(2*i/3). - Wesley Ivan Hurt, May 22 2017
a(-n) = a(n). - Paul Curtz, Jan 19 2020
a(n) = A001399(2*n - 3). - Gus Wiseman, Oct 07 2020
a(n) = (1/6)*(2*n^2 - 3 + gcd(n,3)). - Ridouane Oudra, Apr 15 2021
E.g.f.: (exp(x)*(-2 + 3*x*(1 + x)) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 24 2022
Sum_{n>=2} (-1)^n/a(n) = Pi/sqrt(3) - Pi^2/36 - 3/4. - Amiram Eldar, Dec 02 2022

Edited by Charles R Greathouse IV, Apr 19 2010

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

A168184 Characteristic function of numbers that are not multiples of 10.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Nov 30 2009

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A168185, A145568, A168182, A168181, A109720, A097325, A011558, A166486, A011655, A000035, A010690, A033442.

a168184 = (1 -) . (0 ^) . (`mod` 10)
a168184_list = cycle [0,1,1,1,1,1,1,1,1,1]
-- Reinhard Zumkeller, Oct 10 2012

Table[If[Mod[n,10]==0,0,1],{n,0,110}] (* or *) PadRight[{},110,{0,1,1,1,1,1,1,1,1,1}] (* Harvey P. Dale, Jun 03 2023 *)

a(n)=n%10>0 \\ Charles R Greathouse IV, Sep 24 2015

a(n+10) = a(n);
a(n) = A000007(A010879(n));
a(A067251(n)) = 1; a(A008592(n)) = 0;
not the same as A168046: a(n)=A168046 for n<=100;
A033442(n) = Sum_{k=0..n} a(k)*(n-k).
Dirichlet g.f.: (1-1/10^s)*zeta(s). - R. J. Mathar, Feb 19 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013

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

A033438 Number of edges in 6-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 20, 26, 33, 41, 50, 60, 70, 81, 93, 106, 120, 135, 150, 166, 183, 201, 220, 240, 260, 281, 303, 326, 350, 375, 400, 426, 453, 481, 510, 540, 570, 601, 633, 666, 700, 735, 770, 806, 843, 881

Offset: 0

N. J. A. Sloane

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,6) (also sequence R_n(5,6)) 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

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

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 [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Differs from A025708(n)+1 at 31st position.
Cf. A002620, A000212, A033436, A033437, A033439, A033440, A033441, A033442, A033443, A033444. [From 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)).

a[n_] := Floor[5n^2/12];
Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Jul 31 2018, after Peter Bala *)

a(n) = Sum_{k=0..n} A097325(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
a(n) = +2*a(n-1) -a(n-2) +a(n-6) -2*a(n-7) +a(n-8).
G.f.: -x^2*(1+x+x^3+x^4+x^2) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^3 ).
a(n) = floor(5*n^2/12). - Peter Bala, Aug 12 2013
a(n) = Sum_{i=1..n} floor(5*i/6). - Wesley Ivan Hurt, Sep 12 2017

A033439 Number of edges in 7-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 27, 34, 42, 51, 61, 72, 84, 96, 109, 123, 138, 154, 171, 189, 207, 226, 246, 267, 289, 312, 336, 360, 385, 411, 438, 466, 495, 525, 555, 586, 618, 651, 685, 720, 756, 792, 829, 867, 906, 946, 987, 1029, 1071, 1114, 1158, 1203, 1249

Offset: 0

N. J. A. Sloane

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,7) (also sequence R_n(6,7)) 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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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 [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033440, A033441, A033442, A033443, A033444. [From 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)).

[Floor(3*n^2/7): n in [0..60]]; // Vincenzo Librandi, Oct 19 2013

CoefficientList[Series[- x^2 (x + 1) (x^2 - x + 1) (x^2 + x + 1)/((x - 1)^3 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,0,1,3,6,10,15,21,27},60] (* Harvey P. Dale, Mar 19 2015 *)

a(n) = Sum_{k=0..n} A109720(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x+1)*(x^2-x+1)*(x^2+x+1)/((x-1)^3*(x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Aug 09 2012]
a(n) = floor(3*n^2/7). - Peter Bala, Aug 12 2013
a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10, a(6)=15, a(7)=21, a(8)=27, a(n)=2*a(n-1)-a(n-2)+a(n-7)-2*a(n-8)+a(n-9). - Harvey P. Dale, Mar 19 2015
a(n) = Sum_{i=1..n} floor(6*i/7). - Wesley Ivan Hurt, Sep 12 2017

More terms from Vincenzo Librandi, Oct 19 2013

A033444 Number of edges in 12-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 77, 89, 102, 116, 131, 147, 164, 182, 201, 221, 242, 264, 286, 309, 333, 358, 384, 411, 439, 468, 498, 529, 561, 594, 627, 661, 696, 732, 769, 807, 846, 886, 927, 969, 1012, 1056, 1100, 1145, 1191, 1238, 1286

Offset: 0

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Turán Graph [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033440, A033441, A033442, A033443. [Reinhard Zumkeller, Nov 30 2009]

CoefficientList[Series[- x^2 (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/((x - 1)^3 (x + 1) (x^2 - x + 1) (x^2 + 1) (x^2 + x + 1) (x^4 - x^2 + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 20 2013 *)

a(n) = Sum_{k=0..n} A168185(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)^3*(x+1)*(x^2-x+1)*(x^2+1)*(x^2+x+1)*(x^4-x^2+1)). [Colin Barker, Aug 09 2012]
a(n) = Sum_{i=1..n} floor(11*i/12). - Wesley Ivan Hurt, Sep 12 2017

More terms from Vincenzo Librandi, Oct 20 2013

A033440 Number of edges in 8-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 35, 43, 52, 62, 73, 85, 98, 112, 126, 141, 157, 174, 192, 211, 231, 252, 273, 295, 318, 342, 367, 393, 420, 448, 476, 505, 535, 566, 598, 631, 665, 700, 735, 771, 808, 846, 885, 925

Offset: 0

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Turán Graph [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033441, A033442, A033443, A033444. [Reinhard Zumkeller, Nov 30 2009]

CoefficientList[Series[- x^2 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/((x - 1)^3 (x + 1) (x^2 + 1) (x^4 + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1},{0,0,1,3,6,10,15,21,28,35},50] (* Harvey P. Dale, Mar 23 2015 *)

a(n) = round( (7/16)*n(n-2) ) +0 or -1 depending on n: if there is k such 8k+4<=n<=8k+6 then a(n) = floor( (7/16)*n*(n-2)) otherwise a(n) = round( (7/16)*n(n-2)). E.g. because 8*2+4<=21<=8*2+6 a(n) = floor((7/16)*21*19) = floor(174, 5625)=174. - Benoit Cloitre, Jan 17 2002
a(n) = Sum_{k=0..n} A168181(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)^3*(x+1)*(x^2+1)*(x^4+1)). [Colin Barker, Aug 09 2012]
a(n) = Sum_{i=1..n} floor(7*i/8). - Wesley Ivan Hurt, Sep 12 2017

cp's OEIS Frontend

A000212 a(n) = floor(n^2/3).

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

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A168184 Characteristic function of numbers that are not multiples of 10.

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

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A033438 Number of edges in 6-partite Turán graph of order n.

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A033439 Number of edges in 7-partite Turán graph of order n.

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A033444 Number of edges in 12-partite Turán graph of order n.

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