A000212 -id:A000212 - cp's OEIS frontend

A158405 Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m

1, 1, 3, 1, 3, 5, 1, 3, 5, 7, 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, 11, 1, 3, 5, 7, 9, 11, 13, 1, 3, 5, 7, 9, 11, 13, 15, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23

Offset: 1

Paul Curtz, Mar 18 2009

Row sums are n^2 = A000290(n).
The triangle sums, see A180662 for their definitions, link this triangle of odd numbers with seventeen different sequences, see the crossrefs. The knight sums Kn14 - Kn110 have been added. - Johannes W. Meijer, Sep 22 2010
A208057 is the eigentriangle of A158405 such that as infinite lower triangular matrices, A158405 * A208057 shifts the latter, deleting the right border of 1's. - Gary W. Adamson, Feb 22 2012
T(n,k) = A099375(n-1,n-k), 1<=k<=n. [Reinhard Zumkeller, Mar 31 2012]

			The triangle contains the first n odd numbers in row n:
  1;
  1,3;
  1,3,5;
  1,3,5,7;
From _Seiichi Manyama_, Dec 02 2017: (Start)
    |       a(n)        |                               | A000290(n)
   -----------------------------------------------------------------
   0|                                                      (=  0)
   1|                 1 = 1/3 * ( 3)                       (=  1)
   2|             1 + 3 = 1/3 * ( 5 +  7)                  (=  4)
   3|         1 + 3 + 5 = 1/3 * ( 7 +  9 + 11)             (=  9)
   4|     1 + 3 + 5 + 7 = 1/3 * ( 9 + 11 + 13 + 15)        (= 16)
   5| 1 + 3 + 5 + 7 + 9 = 1/3 * (11 + 13 + 15 + 17 + 19)   (= 25)
(End)

Seiichi Manyama, Rows n = 1..140, flattened
Daniel Erman, The Josephus Problem, Numberphile video (2016)
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Index entries for sequences related to the Josephus Problem

Cf. A129326, A099375, A005408.
Triangle sums (see the comments): A000290 (Row1; Kn11 & Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A000027 (Row2); A005563 (Kn12); A028347 (Kn13); A028560 (Kn14); A028566 (Kn15); A098603 (Kn16); A098847 (Kn17); A098848 (Kn18); A098849 (Kn19); A098850 (Kn110); A000217 (Kn21. Kn22, Kn23, Fi2, Ze2); A000384 (Kn3, Fi1, Ze3); A000212 (Ca2 & Ze4); A000567 (Ca3, Ze1); A011848 (Gi2); A001107 (Gi3). - Johannes W. Meijer, Sep 22 2010
Cf. A063656, A063657, A208057, A292610, A292611.

a158405 n k = a158405_row n !! (k-1)
a158405_row n = a158405_tabl !! (n-1)
a158405_tabl = map reverse a099375_tabl
-- Reinhard Zumkeller, Mar 31 2012

Table[2 Range[1, n] - 1, {n, 12}] // Flatten (* Michael De Vlieger, Oct 01 2015 *)

a(n) = 2*(n-floor((-1+sqrt(8*n-7))/2)*(floor((-1+sqrt(8*n-7))/2)+1)/2)-1;
vector(100, n, a(n)) \\ Altug Alkan, Oct 01 2015

a(n) = 2*i-1, where i = n-t(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 03 2013

a(n) = 2*A002262(n-1) + 1. - Eric Werley, Sep 30 2015

Edited by R. J. Mathar, Oct 06 2009

A007980 Expansion of (1+x^2)/((1-x)^2*(1-x^3)).

Original entry on oeis.org

1, 2, 4, 7, 10, 14, 19, 24, 30, 37, 44, 52, 61, 70, 80, 91, 102, 114, 127, 140, 154, 169, 184, 200, 217, 234, 252, 271, 290, 310, 331, 352, 374, 397, 420, 444, 469, 494, 520, 547, 574, 602, 631, 660, 690, 721, 752, 784, 817, 850, 884, 919, 954, 990, 1027, 1064

Offset: 0

N. J. A. Sloane

Molien series for ternary self-dual codes over GF(3) of length 12n containing 11...1.
(1+x)*(1+x^2) / ((1-x)*(1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(O_3(q); F_2).
a(n) is the position of the n-th triangular number in the running sum of the (pseudo-Orloj) sequence 1,2,1,2,1,2,1...., cf. A028355. - Wouter Meeussen, Mar 10 2002
a(n) = [a(n-1) + (number of even terms so far in the sequence)]. Example: 14 is [10 + 4 even terms so far in the sequence (they are 0,2,4,10)]. See A096777 for the same construction with odd integers. - Eric Angelini, Aug 05 2007
The number of partitions of 2*n into at most 3 parts. - Colin Barker, Mar 31 2015
Also a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of a trigonal Rotational Energy Surface. An optimal basis for the expansion follows either decomposition: g1(x) = (1+x)(1+x^2)g2(x) or g1(x) = (1+x^2)x^(-1)g3(x), where g1(x), g2(x), g3(x) are the generating functions for sequences A007980, A001399, A001840. - Bradley Klee, Aug 06 2015
Also a(n) equals the number of linearly-independent terms at 4n-th order in the power series expansion of the symmetrized weight enumerator of a self-dual code of length n over Z4 that contains a vector (+/-)1^n and has all norms divisible by 8. An optimal basis for the expansion follows the decomposition: g1(x) = (1+x)(1+x^2)g2(x) where g1(x), g2(x) are the generating functions for sequences A007980, A001399. (Cf. Calderbank and Sloane, Corollary 5.) - Bradley Klee, Aug 06 2015
Also, a(n) is equal to the number of partitions of 2n+3 of length 3. Letting n=4, there are a(4)=10 partitions of 2n+3=11 of length 3: (9,1,1), (8,2,1), (7,3,1), (7,2,2), (6,4,1), (6,3,2), (5,5,1), (5,4,2), (5,3,3), (4,4,3). - John M. Campbell, Jan 30 2016
a(n) is the number of partitions of n into parts 1 (of two kinds), part 2 (occurring at most once), and parts 3. - Joerg Arndt, Oct 12 2020
Conjecture: a(n) is the maximum number of pieces a triangle can be cut into by n cevians. - Anton Zakharov, Apr 04 2017
Also, a(n) is the number of graphs which are double-triangle descendants of K_5 with n+6 triangles and 3 more vertices than triangles. See Laradji/Mishna/Yeats reference, proposition 3.6 for details. - Karen A. Yeats, Feb 21 2020

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 10*x^4 + 14*x^5 + 19*x^6 + 24*x^7 + ...

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 233.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (Abstract, pdf, ps).
Mohamed Laradji, Marni Mishna, and Karen Yeats, Some results on double triangle descendants of K_5, arXiv:1904.06923 [math.CO], 2019.
C. L. Mallows and N. J. A. Sloane, Weight enumerators of self-orthogonal codes over GF(3), SIAM J. Alg. Discrete Methods, 2 (1981), 452-460.
Paul Tabatabai and Dieter P. Gruber, Knights and Liars on Graphs, J. Int. Seq., Vol. 24 (2021), Article 21.5.8.
Anton Zakharov, cevians.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Index entries for Molien series.
Index entries for two-way infinite sequences.

Cf. A000326, A001399, A001840, A002378, A003215, A004396, A005449, A007980.

Cf. A008796, A028355, A049450, A069905, A096777, A192736, A211540, A238705.

with (combinat):seq(count(Partition((2*n+1)), size=3), n=1..56); # Zerinvary Lajos, Mar 28 2008

Table[Ceiling[n (n+1)/3], {n, 56}]
CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^3)),{x,0,60}],x] (* Vincenzo Librandi, Feb 25 2012 *)
a[ n_] := Quotient[ n^2, 3] + n + 1; (* Michael Somos, Aug 23 2015 *)
LinearRecurrence[{2,-1,1,-2,1},{1,2,4,7,10},60] (* Harvey P. Dale, Aug 24 2016 *)

{a(n) = if( n<-1, a(-3-n), polcoeff( (1 + x^2) / ( (1 - x)^2 * (1 - x^3)) + x*O(x^n), n))}; /* Michael Somos, Jun 07 2003 */

{a(n) = n^2\3 + n+1}; /* Michael Somos, Aug 23 2015 */

a(n) = #partitions(2*n, ,[1,3]); \\ Michel Marcus, Feb 12 2016

a(n) = #partitions(2*n+3, ,[3,3]); \\ Michel Marcus, Feb 12 2016

G.f.: (1 + x^2) / ((1 - x)^2 * (1 - x^3)). - Michael Somos, Jun 07 2003
a(n) = a(n-1) + a(n-3) -a(n-4) + 2 = a(-3-n) for all n in Z. - Michael Somos, Jun 07 2003
a(n) = ceiling((n+1)*(n+2)/3). - Paul Boddington, Jan 26 2004
a(n) = A192736(n+1) / (n+1). - Reinhard Zumkeller, Jul 08 2011
From Bruno Berselli, Oct 22 2010: (Start)
a(n) = ((n+1)*(n+2)+(2*cos(2*Pi*n/3)+1)/3)/3 = Sum_{i=1..n+1} A004396(i).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4.
a(n) = A002378(n+1)/3 if 3 divides A002378(n+1), a(n) = (A002378(n)+1)/3 otherwise. (End)
a(n) = A001840(n+1) + A001840(n-1). - R. J. Mathar, Aug 23 2015
From Michael Somos, Aug 23 2015: (Start)
Euler transform of length 4 sequence [2, 1, 1, -1].
a(n) = A001399(2*n) = A008796(2*n) = A008796(2*n + 3) = A069905(2*n + 3) = A211540(2*n + 5).
a(2*n)     = A238705(n+1).
a(3*n - 1) = A049451(n).
a(3*n)     = A003215(n).
a(3*n + 1) = A049450(n+1).
2*a(3*n - 1)  = A005449(n).
2*a(3*n + 1)  = A000326(n+1).
a(n+1) - a(n) = A004396(n+2). (End)
a(n) = floor((n^2+3*n+3)/3). - Giacomo Guglieri, May 01 2019
a(n) = A000212(n) + n+1. - Yuchun Ji, Oct 12 2020
Sum_{n>=0} 1/a(n) = (tanh(Pi/(2*sqrt(3)))-1)*Pi/sqrt(3) + 3. - Amiram Eldar, May 20 2023

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

A337599 Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 9, 5, 10, 0, 16, 2, 14, 7, 17, 0, 27, 1, 21, 11, 24, 6, 36, 1, 30, 15, 37, 2, 51, 1, 41, 25, 44, 2, 64, 5, 58, 25, 57, 2, 81, 13, 69, 31, 70, 3, 108, 5, 80, 43, 85, 17, 123, 5, 97, 46, 120, 6, 144, 6

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A082024 at a(31) = 1, A082024(31) = 0.

The first relatively prime triple is (15,10,6), counted under a(31).

			The a(6) = 1 through a(16) = 5 partitions are (empty columns indicated by dots, A..G = 10..16):
  222  .  422  333  442  .  444  .  644  555  664  .  666  .  866
                    622     633     662  663  844     864     884
                            642     842  933  862     882     A55
                            822     A22       A42     963     A64
                                              C22     A44     A82
                                                      A62     C44
                                                      C33     C62
                                                      C42     E42
                                                      E22     G22

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A014612 intersected with A337694 ranks these partitions.
A200976 and A328673 count these partitions of any length.
A284825 is the case that is also relatively prime.
A307719 is the pairwise coprime instead of non-coprime version.
A335402 gives the positions of zeros.
A337604 is the ordered version.
A337605 is the strict case.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts strict pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000212, A000217, A001840, A018783, A082024, A211540, A220377, A337461.

stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Table[Length[Select[IntegerPartitions[n,{3}],stabQ[#,CoprimeQ]&]],{n,0,100}]

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

A173196 Partial sums of A002620.

Original entry on oeis.org

0, 0, 1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715, 825, 946, 1078, 1222, 1378, 1547, 1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843, 8372, 8924, 9500

Offset: 0

Jonathan Vos Post, Feb 12 2010

Essentially a duplicate of A002623: 0, 0, followed by A002623.
The only primes in this sequence are 3, 7, and 13: for n > 2 both a(2*n+1) = n*(n+1)*(4*n+5)/6 and a(2*n) = n*(n+1)*(4*n-1)/6 are composite. - Bruno Berselli, Jan 19 2011
a(n-1) is the number of integer-sided scalene triangles with largest side <= n, including degenerate (i.e., collinear) triangles. a(n-2) is the number of non-degenerate integer-sided scalene triangles. - Alexander Evnin, Oct 12 2010
Also n-th differences of square pyramidal numbers (A000330) and numbers of triangles in triangular matchstick arrangement of side n (A002717). - Konstantin P. Lakov, Apr 13 2018
Also the number of undirected bishop moves on a n X n chessboard, counted up to rotations and reflections of the board. - Hilko Koning, Aug 16 2025

			a(57) = 0 + 0 + 1 + 2 + 4 + 6 + 9 + 12 + 16 + 20 + 25 + 30 + 36 + 42 + 49 + 56 + 64 + 72 + 81 + 90 + 100 + 110 + 121 + 132 + 144 + 156 + 169 + 182 + 196 + 210 + 225 + 240 + 256 + 272 + 289 + 306 + 324 + 342 + 361 + 380 + 400 + 420 + 441 + 462 + 484 + 506 + 529 + 552 + 576 + 600 + 625 + 650 + 676 + 702 + 729 + 756 + 784 + 812 = 15834.

A. Yu. Evnin. Problem book on discrete mathematics. Moscow: Librokom, 2010; problem 787. (In Russian)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
W. Lanssens, B. Demoen, and P.-L. Nguyen, The Diagonal Latin Tableau and the Redundancy of its Disequalities, Report CW 666, July 2014, Department of Computer Science, KU Leuven.
M. Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Int. Seq. 14 (2011) # 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A000212, A000330, A002620, A002623, A002717, A002984, A007590, A024206, A056827, A072280, A087811, A118013, A118015.

[Floor((2*n^3+3*n^2-2*n)/24): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

CoefficientList[Series[x^2/((1 - x)^3 (1 - x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
Accumulate[Floor[Range[0,60]^2/4]] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,0,1,3,7},60] (* Harvey P. Dale, Feb 09 2020 *)
a[ n_] := Quotient[2 n^3 + 3 n^2 - 2 n, 24]; (* Michael Somos, Jan 14 2021 *)

G.f.: x^2 / ((1-x)^3 * (1-x^2)).
a(n) = (4*n^3 + 6*n^2 - 4*n - 3 + 3*(-1)^n)/48. - Bruno Berselli, Jan 19 2011
a(n) = A002623(n-2) for n >= 2. - Martin von Gagern, Dec 05 2014
a(n) = Sum_{i=0..n} A002620(i) = Sum_{i=0..n} floor(i/2)*ceiling(i/2) = Sum_{i=0..n} floor(i^2/4).
a(n) = round((2*n^3 + 3*n^2 - 2*n)/24) = round((4*n^3 + 6*n^2 - 4*n - 3)/48) = floor((2*n^3 + 3*n^2 - 2*n)/24) = ceiling((2*n^3 + 3*n^2 - 2*n - 3)/24). - Mircea Merca, Nov 23 2010
a(n) = a(n-2) + n*(n-1)/2, n > 1. - Mircea Merca, Nov 25 2010
a(n) = floor(n/2)*(floor(n/2)+1)*(8*ceiling(n/2) - 2*n - 1)/6. - Alexander Evnin, Oct 12 2010
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Jan 14 2021
E.g.f.: (x*(3 + 9*x + 2*x^2)*cosh(x) - (3 - 3*x - 9*x^2 - 2*x^3)*sinh(x))/24. - Stefano Spezia, Jun 02 2021

A056865 a(n) = floor(n^2/10).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 25, 28, 32, 36, 40, 44, 48, 52, 57, 62, 67, 72, 78, 84, 90, 96, 102, 108, 115, 122, 129, 136, 144, 152, 160, 168, 176, 184, 193, 202, 211, 220, 230, 240, 250, 260, 270, 280, 291, 302, 313, 324

Offset: 0

N. J. A. Sloane, Sep 02 2000

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

Cf. A056864.

Cf. A000290, A007590, A000212, A002620, A118015, A056827, A056834, A130519, A056838.

A056865 := proc(n)
    floor(n^2/10) ;
end proc:
seq(A056865(n),n=0..100) ; # R. J. Mathar, Mar 08 2016

Floor[Range[0,60]^2/10] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,1,2,3,4,6,8,10,12},60] (* Harvey P. Dale, Jun 29 2022 *)

a(n) = n^2\10; \\ Michel Marcus, Mar 08 2016

G.f.: -x^4*(1+x^4) / ( (1+x)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)*(x-1)^3 ). - R. J. Mathar, Mar 08 2016

A337453 Numbers k such that the k-th composition in standard order is an ordered triple of distinct positive integers.

Original entry on oeis.org

37, 38, 41, 44, 50, 52, 69, 70, 81, 88, 98, 104, 133, 134, 137, 140, 145, 152, 161, 176, 194, 196, 200, 208, 261, 262, 265, 268, 274, 276, 289, 290, 296, 304, 321, 324, 328, 352, 386, 388, 400, 416, 517, 518, 521, 524, 529, 530, 532, 536, 545, 560, 577, 578

Offset: 1

Gus Wiseman, Sep 07 2020

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding triples begins:
     37: (3,2,1)    140: (4,1,3)    289: (3,5,1)
     38: (3,1,2)    145: (3,4,1)    290: (3,4,2)
     41: (2,3,1)    152: (3,1,4)    296: (3,2,4)
     44: (2,1,3)    161: (2,5,1)    304: (3,1,5)
     50: (1,3,2)    176: (2,1,5)    321: (2,6,1)
     52: (1,2,3)    194: (1,5,2)    324: (2,4,3)
     69: (4,2,1)    196: (1,4,3)    328: (2,3,4)
     70: (4,1,2)    200: (1,3,4)    352: (2,1,6)
     81: (2,4,1)    208: (1,2,5)    386: (1,6,2)
     88: (2,1,4)    261: (6,2,1)    388: (1,5,3)
     98: (1,4,2)    262: (6,1,2)    400: (1,3,5)
    104: (1,2,4)    265: (5,3,1)    416: (1,2,6)
    133: (5,2,1)    268: (5,1,3)    517: (7,2,1)
    134: (5,1,2)    274: (4,3,2)    518: (7,1,2)
    137: (4,3,1)    276: (4,2,3)    521: (6,3,1)

6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts these compositions.
A007304 is an unordered version.
A014311 is the non-strict version.
A337461 counts the coprime case.
A000217(n - 2) counts 3-part compositions.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.
A014612 ranks 3-part partitions.
Cf. A000212, A220377, A307534, A337459, A337460, A337561, A337603, A337604.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Length[stc[#]]==3&&UnsameQ@@stc[#]&]

These triples are counted by 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1).

Intersection of A014311 and A233564.

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

cp's OEIS Frontend

A158405 Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m

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A007980 Expansion of (1+x^2)/((1-x)^2*(1-x^3)).

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

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A337599 Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.

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

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

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A173196 Partial sums of A002620.

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