A000212 -id:A000212 - cp's OEIS frontend

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

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

A184535 a(n) = floor(3/5 * n^2), with a(1)=1.

Original entry on oeis.org

1, 2, 5, 9, 15, 21, 29, 38, 48, 60, 72, 86, 101, 117, 135, 153, 173, 194, 216, 240, 264, 290, 317, 345, 375, 405, 437, 470, 504, 540, 576, 614, 653, 693, 735, 777, 821, 866, 912, 960, 1008, 1058, 1109, 1161, 1215, 1269, 1325, 1382, 1440, 1500, 1560, 1622, 1685, 1749, 1815, 1881, 1949, 2018, 2088, 2160, 2232, 2306, 2381

Offset: 1

Clark Kimberling, Jan 16 2011

Apart from the initial term this is the elliptic troublemaker sequence R_n(2,5) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences see the cross references below. - Peter Bala, Aug 08 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
K. E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051v3 [math.NT]
Wikipedia, Maximum number of deciamonds in a hexagon.
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 1, -2, 1).

Cf. A183532, A279169.

Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A002620 (= R_n(1,2)), A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), 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)), A184535 (= R_n(2,5) = R_n(3,5)).

Concatenation([1], List([2..10^3], n->Int(3/5 * n^2))); # Muniru A Asiru, Feb 04 2018

1,seq(floor(3/5*n^2), n=2..10^3); # Muniru A Asiru, Feb 04 2018

p[n_] := FractionalPart[(n^3 + 5)^(1/3)]; q[n_] := Floor[1/p[n]]; Table[q[n], {n, 1, 120}]
Join[{1},LinearRecurrence[{2, -1, 0, 0, 1, -2, 1},{2, 5, 9, 15, 21, 29, 38},62]] (* Ray Chandler, Aug 31 2015 *)

a(n) = if(n==1, 1, 3*n^2\5); \\ Altug Alkan, Mar 03 2018

def A184535(n): return 3*n**2//5 if n>1 else 1 # Chai Wah Wu, Aug 04 2025

a(n) = floor(1/{(5+n^3)^(1/3)}), where {}=fractional part.
a(n)= +2*a(n-1) -a(n-2) +a(n-5) -2*a(n-6) +a(n-7), for n>8, with g.f. 1-x^2*(1+x)*(2*x^2-x+2)/ ((x^4+x^3+x^2+x+1) *(x-1)^3), so a(n) is (3n^2-2)/5 plus a fifth of A164116 for n>1. [Bruno Berselli, Jan 30 2011. See the following Bala's comment.]
From Peter Bala, Aug 08 2013: (Start)
a(n) = floor(3/5*n^2) for n >= 2.
The sequence b(n) := floor(3/5*n^2) - 3/5*n^2, n >= 1, is periodic with period [-3/5, -2/5, -2/5, -3/5, 0] of length 5. The generating function and recurrence equation given above easily follow from these observations.
The sequence c(n) := 5/2*( (2*n/5 - floor(2*n/5))^2 - (2*n/5 - floor(2*n/5)) ) is also periodic with period 5, and calculation shows it has the same period as the sequence b(n). Thus b(n) = c(n), yielding the alternative formula a(n) = 3/5*n^2 + 5/2*( (2*n/5 - floor(2*n/5))^2 - (2*n/5 - floor(2*n/5)) ), which is one of the formulas for the elliptic troublemaker sequence R_n(2,5) given in Stange (see Section 7, equation (21)). (End)

Better name from Peter Bala, Aug 08 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

A056834 a(n) = floor(n^2/7).

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 7, 9, 11, 14, 17, 20, 24, 28, 32, 36, 41, 46, 51, 57, 63, 69, 75, 82, 89, 96, 104, 112, 120, 128, 137, 146, 155, 165, 175, 185, 195, 206, 217, 228, 240, 252, 264, 276, 289, 302, 315, 329, 343, 357, 371, 386, 401, 416, 432, 448

Offset: 0

N. J. A. Sloane, Sep 02 2000

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

Cf. A000290, A007590, A000212, A002620, A118015, A056827, A130519, A056838, A056865.

Floor[(Range[0,60]^2)/7] (* or *) LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,0,0,1,2,3,5,7,9},60] (* Harvey P. Dale, Jul 21 2014 *)
CoefficientList[Series[-x^3 (1 + x) (x^2 - x + 1)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (x - 1)^3), {x, 0, 100}], x] (* Vincenzo Librandi, Jul 22 2014 *)

a(n) = n^2\7; \\ Michel Marcus, Mar 03 2022

a(n) = +2*a(n-1) -a(n-2) +a(n-7) -2*a(n-8) +a(n-9).

G.f.: -x^3*(1+x)*(x^2-x+1) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^3 ).

A229154 The clubs patterns appearing in n X n coins, with rotation allowed.

Original entry on oeis.org

1, 2, 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, 972, 1008, 1045

Offset: 2

Kival Ngaokrajang, Sep 15 2013

On the Japanese TV show "Tsuki no Koibito", a girl told her boyfriend that she saw a heart in 4 coins. Actually there are a total of 6 distinct patterns appearing in 2 X 2 coins in which each pattern consists of a part of the perimeter of each coin and forms a continuous area.

a(n) is the number of clubs patterns appearing in n X n coins with rotation allowed. It is also A000212, except for the fourth term. The number of inverse patterns (stars or voids between clubs) is A143978 (except for the first term).

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000212, A143978, A074148 (Heart patterns), A227906, A229093 (Clubs pattern, fixed Orientation).

CoefficientList[Series[-(x^6 - 2 x^5 + x^4 - x^3 + 2 x^2 + 1)/((x - 1)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 08 2013 *)

Vec(-x^2*(x^6-2*x^5+x^4-x^3+2*x^2+1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 08 2013

a(n) = floor(n^2/3), a(3) = 2.
From Colin Barker, Oct 08 2013: (Start)
a(n) = n^2/3 + (2/9)*cos((2*Pi*n)/3) - 2/9.
G.f.: -x^2*(x^6-2*x^5+x^4-x^3+2*x^2+1) / ((x-1)^3*(x^2+x+1)). (End)

More terms from Colin Barker, Oct 08 2013

A337484 Number of ordered triples of positive integers summing to n that are neither strictly increasing nor strictly decreasing.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 8, 13, 17, 22, 28, 35, 41, 50, 58, 67, 77, 88, 98, 111, 123, 136, 150, 165, 179, 196, 212, 229, 247, 266, 284, 305, 325, 346, 368, 391, 413, 438, 462, 487, 513, 540, 566, 595, 623, 652, 682, 713, 743, 776, 808, 841, 875, 910, 944, 981, 1017

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

			The a(3) = 1 through a(7) = 13 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)
           (1,2,1)  (1,2,2)  (1,3,2)  (1,3,3)
           (2,1,1)  (1,3,1)  (1,4,1)  (1,4,2)
                    (2,1,2)  (2,1,3)  (1,5,1)
                    (2,2,1)  (2,2,2)  (2,1,4)
                    (3,1,1)  (2,3,1)  (2,2,3)
                             (3,1,2)  (2,3,2)
                             (4,1,1)  (2,4,1)
                                      (3,1,3)
                                      (3,2,2)
                                      (3,3,1)
                                      (4,1,2)
                                      (5,1,1)

A140106 is the unordered case.
A242771 allows strictly increasing but not strictly decreasing triples.
A337481 counts these compositions of any length.
A001399(n - 6) counts unordered strict triples.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A069905 counts unordered triples.
A218004 counts strictly increasing or weakly decreasing compositions.
A332745 counts partitions with weakly increasing or weakly decreasing run-lengths.
A332835 counts compositions with weakly increasing or weakly decreasing run-lengths.
A337483 counts triples either weakly increasing or weakly decreasing.
Cf. A000212, A000217, A001840, A014311, A046691, A128422, A156040, A332834, A337461, A337482, A337561, A337603, A337604.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&&!Greater@@#&]],{n,0,15}]

a(n) = 2*A242771(n - 1) - A000217(n - 1), n > 0.
2*A001399(n - 6) = 2*A069905(n - 3) = 2*A211540(n - 1) is the complement.
4*A001399(n - 6) = 4*A069905(n - 3) = 4*A211540(n - 1) is the strict case.
Conjectures from Colin Barker, Sep 13 2020: (Start)
G.f.: x^3*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>6.
(End)

A033441 Number of edges in 9-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 44, 53, 63, 74, 86, 99, 113, 128, 144, 160, 177, 195, 214, 234, 255, 277, 300, 324, 348, 373, 399, 426, 454, 483, 513, 544, 576, 608, 641, 675, 710, 746, 783, 821, 860, 900, 940, 981, 1023, 1066, 1110, 1155, 1201, 1248, 1296

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
Christian Meyer, On conjecture no. 76 arising from the OEIS, preprint, 2004. [cached copy]
Ralf Stephan, Prove or disprove: 100 conjectures from the OEIS, arXiv:math/0409509 [math.CO], 2004.
Eric Weisstein's World of Mathematics, Turán Graph [From _Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [From _Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436 - A033444. - Reinhard Zumkeller, Nov 30 2009

CoefficientList[Series[- x^2 (x + 1) (x^2 + 1) (x^4 + 1)/((x - 1)^3 (x^2 + x + 1) (x^6 + x^3 + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 20 2013 *)
LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 0, 1, -2, 1},{0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 44},55] (* Ray Chandler, Aug 04 2015 *)

G.f.: x*(1/(1-x) - 1/(1-x^9))/(1-x)^2. - Ralf Stephan, Mar 05 2004
a(n) = Sum_{k=0..n} A168182(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
G.f.: -x^2*(x+1)*(x^2+1)*(x^4+1)/((x-1)^3*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Aug 09 2012
a(n) = Sum_{i=1..n} floor(8*i/9). - Wesley Ivan Hurt, Sep 12 2017

More terms from Vincenzo Librandi, Oct 20 2013

A056838 a(n) = floor(n^2/9).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 7, 9, 11, 13, 16, 18, 21, 25, 28, 32, 36, 40, 44, 49, 53, 58, 64, 69, 75, 81, 87, 93, 100, 106, 113, 121, 128, 136, 144, 152, 160, 169, 177, 186, 196, 205, 215, 225, 235, 245, 256, 266, 277, 289, 300, 312, 324, 336, 348

Offset: 0

N. J. A. Sloane, Sep 02 2000

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

Cf. A008740.

Cf. A000290, A007590, A000212, A002620, A118015, A056827, A056834, A130519, A056865.

Floor[Range[0, 100]^2/9] (* Paolo Xausa, Aug 21 2024 *)

G.f.: x^3*(1+x)*(x^2-x+1)^2/((1-x)^3*(1+x+x^2)(x^6+x^3+1)). [R. J. Mathar, Jan 05 2009]

A337483 Number of ordered triples of positive integers summing to n that are either weakly increasing or weakly decreasing.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 5, 8, 10, 13, 16, 20, 23, 28, 32, 37, 42, 48, 53, 60, 66, 73, 80, 88, 95, 104, 112, 121, 130, 140, 149, 160, 170, 181, 192, 204, 215, 228, 240, 253, 266, 280, 293, 308, 322, 337, 352, 368, 383, 400, 416, 433, 450, 468, 485, 504, 522, 541, 560

Offset: 0

Gus Wiseman, Sep 07 2020

			The a(3) = 1 through a(8) = 10 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (2,1,1)  (1,2,2)  (1,2,3)  (1,2,4)  (1,2,5)
                    (2,2,1)  (2,2,2)  (1,3,3)  (1,3,4)
                    (3,1,1)  (3,2,1)  (2,2,3)  (2,2,4)
                             (4,1,1)  (3,2,2)  (2,3,3)
                                      (3,3,1)  (3,3,2)
                                      (4,2,1)  (4,2,2)
                                      (5,1,1)  (4,3,1)
                                               (5,2,1)
                                               (6,1,1)

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

A001399(n - 3) = A069905(n) = A211540(n + 2) counts the unordered case.
2*A001399(n - 6) = 2*A069905(n - 3) = 2*A211540(n - 1) counts the strict case.
A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts the strict unordered case.
A329398 counts these compositions of any length.
A218004 counts strictly increasing or weakly decreasing compositions.
A337484 counts neither strictly increasing nor strictly decreasing compositions.
Cf. A000212, A000217, A001840, A014311, A156040, A337461, A337603, A337604.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],LessEqual@@#||GreaterEqual@@#&]],{n,0,30}]

a(n > 0) = 2*A001399(n - 3) - A079978(n).
From Colin Barker, Sep 08 2020: (Start)
G.f.: x^3*(1 + x + x^2 - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>6. (End)
E.g.f.: (36 - 9*exp(-x) + exp(x)*(6*x^2 + 6*x - 19) - 8*exp(-x/2)*cos(sqrt(3)*x/2))/36. - Stefano Spezia, Apr 05 2023

A337602 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 10, 9, 18, 16, 24, 21, 43, 24, 51, 31, 54, 42, 94, 45, 102, 55, 99, 69, 163, 66, 150, 88, 168, 96, 265, 93, 228, 121, 246, 126, 337, 132, 315, 169, 342, 162, 487, 165, 420, 217, 411, 213, 619, 207, 558, 259, 540, 258, 784, 264, 654, 325, 660

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

			The a(3) = 1 through a(8) = 18 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,3,3)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,5,1)  (1,3,4)
                    (2,1,2)  (1,4,1)  (2,2,3)  (1,4,3)
                    (2,2,1)  (2,1,3)  (2,3,2)  (1,5,2)
                    (3,1,1)  (2,2,2)  (3,1,3)  (1,6,1)
                             (2,3,1)  (3,2,2)  (2,1,5)
                             (3,1,2)  (3,3,1)  (2,3,3)
                             (3,2,1)  (5,1,1)  (2,5,1)
                             (4,1,1)           (3,1,4)
                                               (3,2,3)
                                               (3,3,2)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)

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

The complement in A014311 of A337695 ranks these compositions.
A220377*6 is the strict case.
A337600 is the unordered version.
A337603 does not consider a singleton to be coprime unless it is (1).
A337664 counts these compositions of any length.
A000740 counts relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
A000217 counts 3-part compositions.
A001399/A069905/A211540 count 3-part partitions.
A023023 counts relatively prime 3-part partitions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime 3-part compositions.
Cf. A000212, A007359, A087087, A284825, A302696, A304709, A304712, A307719, A328673, A335235, A335238, A337483, A337562, A337601.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,100}]

cp's OEIS Frontend

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

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A184535 a(n) = floor(3/5 * n^2), with a(1)=1.

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GAP

Maple

Mathematica

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

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Mathematica

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A056834 a(n) = floor(n^2/7).

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Mathematica

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A229154 The clubs patterns appearing in n X n coins, with rotation allowed.

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Mathematica

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A337484 Number of ordered triples of positive integers summing to n that are neither strictly increasing nor strictly decreasing.

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Mathematica

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A033441 Number of edges in 9-partite Turán graph of order n.

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Mathematica