A001840 -id:A001840 - cp's OEIS frontend

A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267

Offset: 0

Clark Kimberling, Jun 01 2012

In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):
  A: 2,  0, -2,  1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);
  B: 3, -2, -2,  3, -1;
  C: 4, -6,  4, -1;
  D: 1,  2, -2, -1,  1;
  E: 2,  1, -4,  1,  2, -1;
  F: 2, -1,  1, -2,  1;
  G: 2, -1,  0,  1, -2,  1;
  H: 2, -1,  2, -4,  2, -1,  2, -1;
  I: 3, -3,  2, -3,  3, -1;
  J: 4, -7,  8, -7,  4, -1.
  ...
A212959 ... |w-x|=|x-y| ...... recurrence type A
A212960 ... |w-x| != |x-y| ................... B
A212683 ... |w-x| < |x-y| .................... B
A212684 ... |w-x| >= |x-y| ................... B
A212963 ... see entry for definition ......... B
A212964 ... |w-x| < |x-y| < |y-w| ............ B
A006331 ... |w-x| < y ........................ C
A005900 ... |w-x| <= y ....................... C
A212965 ... w = R ............................ D
A212966 ... 2*w = R
A212967 ... w < R ............................ E
A212968 ... w >= R ........................... E
A077043 ... w = x > R ........................ A
A212969 ... w != x and x > R ................. E
A212970 ... w != x and x < R ................. E
A055998 ... w = x + y - 1
A011934 ... w < floor((x+y)/2) ............... B
A182260 ... w > floor((x+y)/2) ............... B
A055232 ... w <= floor((x+y)/2) .............. B
A011934 ... w >= floor((x+y)/2) .............. B
A212971 ... w < floor((x+y)/3) ............... B
A212972 ... w >= floor((x+y)/3) .............. B
A212973 ... w <= floor((x+y)/3) .............. B
A212974 ... w > floor((x+y)/3) ............... B
A212975 ... R is even ........................ E
A212976 ... R is odd ......................... E
A212978 ... R = 2*n - w - x
A212979 ... R = average{w,x,y}
A212980 ... w < x + y and x < y .............. B
A212981 ... w <= x+y and x < y ............... B
A212982 ... w < x + y and x <= y ............. B
A212983 ... w <= x + y and x <= y ............ B
A002623 ... w >= x + y and x <= y ............ B
A087811 ... w = 2*x + y ...................... A
A008805 ... w = 2*x + 2*y .................... D
A000982 ... 2*w = x + y ...................... F
A001318 ... 2*w = 2*x + y .................... F
A001840 ... w = 3*x + y
A212984 ... 3*w = x + y
A212985 ... 3*w = 3*x + y
A001399 ... w = 2*x + 3*y
A212986 ... 2*w = 3*x + y
A008810 ... 3*x = 2*x + y .................... F
A212987 ... 3*w = 2*x + 2*y
A001972 ... w = 4*x + y ...................... G
A212988 ... 4*w = x + y ...................... G
A212989 ... 4*w = 4*x + y
A008812 ... 5*w = 2*x + 3*y
A016061 ... n < w + x + y <= 2*n ............. C
A000292 ... w + x + y <=n .................... C
A000292 ... 2*n < w + x + y <= 3*n ........... C
A212977 ... n/2 < w + x + y <= n
A143785 ... w < R < x ........................ E
A005996 ... w < R <= x ....................... E
A128624 ... w <= R <= x ...................... E
A213041 ... R = 2*|w - x| .................... A
A213045 ... R < 2*|w - x| .................... B
A087035 ... R >= 2*|w - x| ................... B
A213388 ... R <= 2*|w - x| ................... B
A171218 ... M < 2*m .......................... B
A213389 ... R < 2|w - x| ..................... E
A213390 ... M >= 2*m ......................... E
A213391 ... 2*M < 3*m ........................ H
A213392 ... 2*M >= 3*m ....................... H
A213393 ... 2*M > 3*m ........................ H
A213391 ... 2*M <= 3*m ....................... H
A047838 ... w = |x + y - w| .................. A
A213396 ... 2*w < |x + y - w| ................ I
A213397 ... 2*w >= |x + y - w| ............... I
A213400 ... w < R < 2*w
A069894 ... min(|w-x|,|x-y|) = 1
A000384 ... max(|w-x|,|x-y|) = |w-y|
A213395 ... max(|w-x|,|x-y|) = w
A213398 ... min(|w-x|,|x-y|) = x ............. A
A213399 ... max(|w-x|,|x-y|) = x ............. D
A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D
A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E
A006918 ... |w-x| + |x-y| > w+x+y ............ E
A213481 ... |w-x| + |x-y| <= w+x+y ........... E
A213482 ... |w-x| + |x-y| < w+x+y ............ E
A213483 ... |w-x| + |x-y| >= w+x+y ........... E
A213484 ... |w-x|+|x-y|+|y-w| = w+x+y
A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J
A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J
A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J
A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J
A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J
A213490 ... w,x,y,|w-x|,|x-y| distinct
A213491 ... w,x,y,|w-x|,|x-y| not distinct
A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct
A213495 ... w = min(|w-x|,|x-y|,|w-y|)
A213492 ... w != min(|w-x|,|x-y|,|w-y|)
A213496 ... x != max(|w-x|,|x-y|)
A213498 ... w != max(|w-x|,|x-y|,|w-y|)
A213497 ... w = min(|w-x|,|x-y|)
A213499 ... w != min(|w-x|,|x-y|)
A213501 ... w != max(|w-x|,|x-y|)
A213502 ... x != min(|w-x|,|x-y|)
...
A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties.  Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.
Partial sums of the numbers congruent to {1,3} mod 6 (see A047241). - Philippe Deléham, Mar 16 2014

			a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).
Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...
a(0) = 1;
a(1) = 1 + 3 = 4;
a(2) = 1 + 3 + 7 = 11;
a(3) = 1 + 3 + 7 + 9 = 20;
a(4) = 1 + 3 + 7 + 9 + 13 = 33;
a(5) = 1 + 3 + 7 + 9 + 13 + 15 = 48; etc. - _Philippe Deléham_, Mar 16 2014

A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A047241, A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]]   (* A212959 *)

a(n)=(6*n^2+8*n+3)\/4 \\ Charles R Greathouse IV, Jul 28 2015

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).
a(n) + A212960(n) = (n+1)^3.
a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
a(n) = 2*A069905(3*(n+1)+2) - 3*(n+1). - Ayoub Saber Rguez, Aug 31 2021

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

Original entry on oeis.org

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

A051168 Triangular array T(h,k) for 0 <= k <= h read by rows: T(h,k) = number of binary Lyndon words with k ones and h-k zeros.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 5, 5, 3, 1, 0, 0, 1, 3, 7, 8, 7, 3, 1, 0, 0, 1, 4, 9, 14, 14, 9, 4, 1, 0, 0, 1, 4, 12, 20, 25, 20, 12, 4, 1, 0, 0, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 0, 0, 1, 5, 18, 40, 66, 75, 66, 40, 18, 5, 1, 0, 0, 1, 6

Offset: 0

Clark Kimberling

T(h,k) is the number of classes of aperiodic binary words of k ones and h-k zeros; words u,v are in the same class if v is a cyclic permutation of u (e.g., u=111000, v=110001) and a word is aperiodic if not a juxtaposition of 2 or more identical subwords.
T(2n, n), T(2n+1, n), T(n, 3) match A022553, A000108, A001840, respectively. Row sums match A001037.
From R. J. Mathar, Jul 31 2008: (Start)
This triangle may also be regarded as the square array A(r,n), the n-th term of the r-th Witt transform of the all-1 sequence, r >= 1, n >= 0, read by antidiagonals:
This array begins as follows:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
0 1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 51 57 63
0 1 2 5 8 14 20 30 40 55 70 91 112 140 168 204 240 285 330
0 1 3 7 14 25 42 66 99 143 200 273 364 476 612 775 969 1197 1463
0 1 3 9 20 42 75 132 212 333 497 728 1026 1428 1932 2583 3384 4389 5598
0 1 4 12 30 66 132 245 429 715 1144 1768 2652 3876 5537 7752 10659 14421 19228
0 1 4 15 40 99 212 429 800 1430 2424 3978 6288 9690 14520 21318 30624 43263 60060
0 1 5 18 55 143 333 715 1430 2700 4862 8398 13995 22610 35530 54477 81719 120175
0 1 5 22 70 200 497 1144 2424 4862 9225 16796 29372 49742 81686 130750 204248
0 1 6 26 91 273 728 1768 3978 8398 16796 32065 58786 104006 178296 297160 482885
0 1 6 30 112 364 1026 2652 6288 13995 29372 58786 112632 208012 371384 643842
0 1 7 35 140 476 1428 3876 9690 22610 49742 104006 208012 400023 742900 1337220
0 1 7 40 168 612 1932 5537 14520 35530 81686 178296 371384 742900 1432613 2674440
...
It is essentially symmetric: A(r,r+i) = A(r,r-i+1).
Some of the diagonals are:
A(r,r+1): A000108
A(r,r): A022553
A(r,r-1): A000108
A(r,r+2): A000150
A(r,r+3): A050181
A(r,r+4): A050182
A(r,r+5): A050183
A(r,r-2): A000150 (End)
Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} A054533(d, v) * binomial((n + k)/d, k/d) = S(k, n, v). This was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 1) = T(n + k, k). - Petros Hadjicostas, Jul 09 2019

			Triangle begins with:
h=0: 1
h=1: 1, 1
h=2: 0, 1, 0
h=3: 0, 1, 1, 0
h=4: 0, 1, 1, 1,  0
h=5: 0, 1, 2, 2,  1,  0
h=6: 0, 1, 2, 3,  2,  1, 0
h=7: 0, 1, 3, 5,  5,  3, 1, 0
h=8: 0, 1, 3, 7,  8,  7, 3, 1, 0
h=9: 0, 1, 4, 9, 14, 14, 9, 4, 1, 0
...
T(6,3) counts classes {111000},{110100},{110010}, each of 6 aperiodic. The class {100100} contains 3 periodic words, counted by T(3,1) as {100}, consisting of 3 aperiodic words 100,010,001.

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Freddy Cachazo and Nick Early, Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes, arXiv:2010.09708 [math.CO], 2020.
A. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233-238; MR1691428 (2000c:13006).
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173-188; MR1719140 (2000j:05009). See p. 174. - _N. J. A. Sloane_, Aug 06 2014
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Romeo Meštrović, Different classes of binary necklaces and a combinatorial method for their enumerations, arXiv:1804.00992 [math.CO], 2018.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
P. Stanica and S. Maitra, Rotation symmetric boolean functions - count and cryptographic properties, Disc. Appl. Math. 156 (2008) 1567-1580; see h_{n,w} in eq. (3).
Index entries for sequences related to Lyndon words

Columns 1-11: A000012, A004526(n-1), A001840(n-4), A006918(n-4), A011795(n-1), A011796(n-6), A011797(n-1), A031164(n-9), A011845, A032168, A032169. See also A000150.

Cf. A047996, A052307, A052314, A092964, A185158, A123223, A124814.

A := proc(r,n) local gf,d,genf; genf := 1/(1-x) ; gf := 0 ; for d in numtheory[divisors](r) do gf := gf + numtheory[mobius](d)*(subs(x= x^d,genf))^(r/d) ; od: gf := expand(gf/r) ; coeftayl(gf,x=0,n) ; end proc:
A051168 := proc(n,k) if n<=1 then 1; elif n=0 or n=k then 0; else A(n-k,k) ; end if;
end proc:
seq(seq(A051168(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Mar 29 2011

Table[If[n===0,1,1/n Plus@@(MoebiusMu[ # ]Binomial[n/#,k/# ]&/@ Divisors[GCD[n,k]])],{n,0,12},{k,0,n}] (* Wouter Meeussen, Jul 20 2008 *)

{T(n, k) = local(A, ps, c); if( k<0 || k>n, 0, if( n==0 && k==0, 1, A = x * O(x^n) + y * O(y^n); ps = 1 - x - y + A; for( m=1, n, for( i=0, m, c = polcoeff( polcoeff(ps, i, x), m-i, y); if( m==n && i==k, break(2), ps *= (1 -y^(m-i) * x^i + A)^c))); -c))} /* Michael Somos, Jul 03 2004 */

T(n,k) = if (n==0, 1, (1/n) * sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d,k/d)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 16 2018

T(h, k) = 1 for (h, k) in {(0, 0), (1, 0), (1, 1)}; T(h, k) = 0 if h >= 2 and k = 0 or k = h. Otherwise, T(h, k) = (1/h)*(C(h, k)-S(h, k)), where S(h, k) = Sum_{d <= 2, d|h, d|k} (h/d)*T(h/d, k/d).
1 - x - y = Product_{i,j} (1 - x^i * y^j)^T(i+j, j) where i >= 0, j >= 0 are not both zero. - Michael Somos, Jul 03 2004
The prime rows are given by (1+x)^p/p with noninteger coefficients rounded to zero. E.g., for h = 2 below, (1 + x)^2/2 = (1 + 2*x + x^2)/2 = 0.5 + x + 0.5*x^2 gives (0,1,0). - Tom Copeland, Oct 21 2014
T(n,k) = (1/n) * Sum_{d | gcd(n,k)} mu(d) * binomial(n/d, k/d), for n > 0. - Andrew Howroyd, Mar 26 2017
From Petros Hadjicostas, Jun 16 2019: (Start)
O.g.f. for column k >= 1: (x^k/k) * Sum_{d|k} mu(d)/(1 - x^d)^(k/d).
Bivariate o.g.f.: Sum_{n,k >= 0} T(n, k)*x^n*y^k = 1 - Sum_{d >= 1} (mu(d)/d) *log(1 - x^d * (1 + y^d)).
(End)

A008620 Positive integers repeated three times.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26

Offset: 0

N. J. A. Sloane

Arises from Gleason's theorem on self-dual codes: the Molien series is 1/((1-x^8)*(1-x^24)) for the weight enumerators of doubly-even binary self-dual codes; also 1/((1-x^4)*(1-x^12)) for ternary self-dual codes.
The number of partitions of n into distinct parts where each part is either a power of two or three times a power of two.
Number of partitions of n into parts 1 or 3. - Reinhard Zumkeller, Aug 15 2011
The dimension of the space of modular forms on Gamma_1(3) of weight n>0 with a(q) the generator of weight 1 and c(q)^3 / 27 the generator of weight 3 where a(), c() are cubic AGM theta functions. - Michael Somos, Apr 01 2015
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
a(n-1) is the minimal number of circles that can be drawn through n points in general position in the plane. - Anton Zakharov, Dec 31 2016
Number of partitions of n into distinct parts from A029744.- R. J. Mathar, Mar 01 2023
Number of representations n=sum_i c_i*2^i with c_i in {0,1,3,4} [Anders]. See A120562 or A309025 for other c_i sets. - R. J. Mathar, Mar 01 2023

G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 12 Exer. 7
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Eq. (14), p. 601 and Theorem 3c, p. 602; also Problem 5 p. 620.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
K. Anders, Counting Non-Standard Binary Representations, JIS vol 19 (2016) #16.3.3 example 1.
E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason's Theorem on Self-Dual Codes, IEEE Trans. Information Theory, IT-18 (1972), 409-414.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 210
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 449
F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seqs., Vol. 7, 2004.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A000027, A001840, A002264, A004016, A004396, A005882, A005928, A008621, A010766, A049347.

Column 3 of A235791.

a008620 = (+ 1) . (`div` 3)
a008620_list = concatMap (replicate 3) [1..]
-- Reinhard Zumkeller, Feb 19 2013, Apr 16 2012, Sep 25 2011

[Floor(n/3)+1: n in [0..80]]; // Vincenzo Librandi, Aug 16 2011

a := func< n | Dimension( ModularForms( Gamma1(3), n))>; /* Michael Somos, Apr 01 2015 */

A008620:=n->floor(n/3)+1; seq(A008620(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Table[Floor[n/3] + 1, {n, 0, 90}] (* Stefan Steinerberger, Apr 02 2006 *)
Table[{n, n, n}, {n, 30}] // Flatten (* Harvey P. Dale, Jan 15 2017 *)
Ceiling[Range[20]/3] (* Eric W. Weisstein, Aug 12 2023 *)
Table[Ceiling[n/3], {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(1 + n - Cos[2 n Pi/3] + Sin[2 n Pi/3]/Sqrt[3])/3, {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(n - ChebyshevU[n, -1/2] + 1)/3, {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 1, 1, 2}, 20] (* Eric W. Weisstein, Aug 12 2023 *)
CoefficientList[Series[1/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 12 2023 *)

PARI
```
a(n)=n\3+1
```

def a(n) : return( dimension_modular_forms( Gamma1(3), n) ); # Michael Somos, Apr 01 2015

a(n) = floor(n/3) + 1.
a(n) = A010766(n+3, 3).
G.f.: 1/((1-x)*(1-x^3)) = 1/((1-x)^2*(1+x+x^2)).
a(n) = A001840(n+1) - A001840(n). - Reinhard Zumkeller, Aug 01 2002
From Paul Barry, May 19 2004: (Start)
Convolution of A049347 and A000027.
a(n) = Sum_{k=0..n} (k+1)*2*sqrt(3)*cos(2*Pi*(n-k)/3 + Pi/6)/3. (End)
The g.f. is 1/(1-V_trefoil(x)), where V_trefoil is the Jones polynomial of the trefoil knot. - Paul Barry, Oct 08 2004
a(2n) = A004396(n+1). - Philippe Deléham, Dec 14 2006
a(n) = ceiling(n/3), n>=1. - Mohammad K. Azarian, May 22 2007
E.g.f.: exp(x)*(2 + x)/3 + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 17 2022

A337461 Number of pairwise coprime ordered triples of positive integers summing to n.

Original entry on oeis.org

0, 0, 0, 1, 3, 3, 9, 3, 15, 9, 21, 9, 39, 9, 45, 21, 45, 21, 87, 21, 93, 39, 87, 39, 153, 39, 135, 63, 153, 57, 255, 51, 207, 93, 225, 93, 321, 81, 291, 135, 321, 105, 471, 105, 393, 183, 381, 147, 597, 147, 531, 213, 507, 183, 759, 207, 621, 273, 621, 231

Offset: 0

Gus Wiseman, Sep 02 2020

nonn

			The a(3) = 1 through a(9) = 9 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)  (1,1,7)
           (1,2,1)  (1,3,1)  (1,2,3)  (1,5,1)  (1,2,5)  (1,3,5)
           (2,1,1)  (3,1,1)  (1,3,2)  (5,1,1)  (1,3,4)  (1,5,3)
                             (1,4,1)           (1,4,3)  (1,7,1)
                             (2,1,3)           (1,5,2)  (3,1,5)
                             (2,3,1)           (1,6,1)  (3,5,1)
                             (3,1,2)           (2,1,5)  (5,1,3)
                             (3,2,1)           (2,5,1)  (5,3,1)
                             (4,1,1)           (3,1,4)  (7,1,1)
                                               (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

A000212 counts the unimodal instead of coprime version.
A220377*6 is the strict case.
A307719 is the unordered version.
A337462 counts these compositions of any length.
A337563 counts the case of partitions with no 1's.
A337603 only requires the *distinct* parts to be pairwise coprime.
A337604 is the intersecting instead of coprime version.
A014612 ranks 3-part partitions.
A302696 ranks pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000217, A001399, A001840, A014311, A101268, A284825, A337562, A326675, A333227, A337601, A337602.

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

A007997 a(n) = ceiling((n-3)(n-4)/6).

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 7, 10, 12, 15, 19, 22, 26, 31, 35, 40, 46, 51, 57, 64, 70, 77, 85, 92, 100, 109, 117, 126, 136, 145, 155, 166, 176, 187, 199, 210, 222, 235, 247, 260, 274, 287, 301, 316, 330, 345, 361, 376, 392, 409, 425, 442, 460, 477, 495, 514, 532, 551, 571, 590, 610

Offset: 3

N. J. A. Sloane

Number of solutions to x+y+z=0 (mod m) with 0<=x<=y<=z

Nonorientable genus of complete graph on n nodes.

Also (with different offset) Molien series for alternating group A_3.

(1+x^3 ) / ((1-x)*(1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(S_6, F_2).

a(n+5) is the number of necklaces with 3 black beads and n white beads.

The g.f./x^5 is Z(C_3,x), the 3-variate cycle index polynomial for the cyclic group C_3, with substitution x[i]->1/(1-x^i), i=1,2,3. Therefore by Polya enumeration a(n+5) is the number of cyclically inequivalent 3-necklaces whose 3 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... . See A102190 for Z(C_3,x). - Wolfdieter Lang, Feb 15 2005

a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x = (y mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012

From Gus Wiseman, Oct 17 2020: (Start)

Also the number of 3-part integer compositions of n - 2 that are either weakly increasing or strictly decreasing. For example, the a(5) = 1 through a(13) = 15 compositions are:

(111) (112) (113) (114) (115) (116) (117) (118) (119)

(122) (123) (124) (125) (126) (127) (128)

(222) (133) (134) (135) (136) (137)

(321) (223) (224) (144) (145) (146)

(421) (233) (225) (226) (155)

(431) (234) (235) (227)

(521) (333) (244) (236)

(432) (334) (245)

(531) (532) (335)

(621) (541) (344)

(631) (542)

(721) (632)

(641)

(731)

(821)

(End)

			For m=7 (n=12), the 12 solutions are xyz = 000 610 520 511 430 421 331 322 662 653 644 554.

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004, p. 204.
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987; see \bar{I}(n) p. 221.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 740.
E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, 2004.

T. D. Noe, Table of n, a(n) for n = 3..1000
Chwas Ahmed, Paul Martin, and Volodymyr Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015-2019.
Alexander Taveira Blomenhofer, On Uniqueness of Power Sum Decomposition, SIAM J. Appl. Alg. Geom. (2025) Vol. 9, Iss. 1.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See pp. 3, 19.
Magnus Rahbek Hansen, Invariant Theory and Graphs, Master's Thesis, Univ. Copenhagen (Denmark, 2023). See p. 38.
Mónica A. Reyes, Cristina Dalfó, Miguel Àngel Fiol, and Arnau Messegué, A general method to find the spectrum and eigenspaces of the k-token of a cycle, and 2-token through continuous fractions, arXiv:2403.20148 [math.CO], 2024. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Index entries for Molien series.
Index entries for sequences related to necklaces.

Cf. A000031, A007998, A003050, A047996, A048259, A053618.
Apart from initial term, same as A058212.
Cf. A000212, A000217, A001840, A128422, A156040.
A001399(n-6)*2 = A069905(n-3)*2 = A211540(n-1)*2 counts the strict case.
A014311 intersected with A225620 U A333256 ranks these compositions.
A218004 counts these compositions of any length.
A000009 counts strictly decreasing compositions.
A000041 counts weakly increasing compositions.
A001523 counts unimodal compositions, with complement counted by A115981.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A333149 counts neither increasing nor decreasing strict compositions.
Cf. A014311, A332834, A333147, A337481, A337482-A337484.

a007997 n = ceiling $ (fromIntegral $ (n - 3) * (n - 4)) / 6
a007997_list = 0 : 0 : 1 : zipWith (+) a007997_list [1..]
-- Reinhard Zumkeller, Dec 18 2013

x^5*(1+x^3)/((1-x)*(1-x^2)*(1-x^3));
seq(ceil(binomial(n,2)/3), n=0..63); # Zerinvary Lajos, Jan 12 2009
a := n -> (n*(n-7)-2*([1,1,-1][n mod 3 +1]-7))/6;
seq(a(n), n=3..64); # Peter Luschny, Jan 13 2015

k = 3; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
Table[Ceiling[((n-3)(n-4))/6],{n,3,100}] (* or *) LinearRecurrence[ {2,-1,1,-2,1},{0,0,1,1,2},100] (* Harvey P. Dale, Jan 21 2014 *)

a(n)=(n^2-7*n+16)\6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = a(n-3) + n - 2, a(0)=0, a(1)=0, a(2)=1 [Offset 0]. - Paul Barry, Jul 14 2004
G.f.: x^5*(1+x^3)/((1-x)*(1-x^2)*(1-x^3)) = x^5*(1-x+x^2)/((1-x)^2*(1-x^3)).
a(n+5) = Sum_{k=0..floor(n/2)} C(n-k,L(k/3)), where L(j/p) is the Legendre symbol of j and p. - Paul Barry, Mar 16 2006
a(3)=0, a(4)=0, a(5)=1, a(6)=1, a(7)=2, a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5). - Harvey P. Dale, Jan 21 2014
a(n) = (n^2 - 7*n + 14 - 2*(-1)^(2^(n + 1 - 3*floor((n+1)/3))))/6. - Luce ETIENNE, Dec 27 2014
a(n) = A001399(n-3) + A001399(n-6). Compare to A140106(n) = A001399(n-3) - A001399(n-6). - Gus Wiseman, Oct 17 2020
a(n) = (40 + 3*(n - 7)*n - 4*cos(2*n*Pi/3) - 4*sqrt(3)*sin(2*n*Pi/3))/18. - Stefano Spezia, Dec 14 2021
Sum_{n>=5} 1/a(n) = 6 - 2*Pi/sqrt(3) + 2*Pi*tanh(sqrt(5/3)*Pi/2)/sqrt(15). - Amiram Eldar, Oct 01 2022

A211540 Number of ordered triples (w,x,y) with all terms in {1..n} and 2w = 3x + 4y.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208, 217, 225, 234, 243, 252, 261, 271, 280, 290

Offset: 0

Clark Kimberling, Apr 15 2012

For a guide to related sequences, see A211422.
Also the number of partitions of n+1 into three parts, where each part > 1. - Peter Woodward, May 25 2015
a(n) is also equal to the number of partitions of n+4 into three distinct parts, where each part > 1. - Giovanni Resta, May 26 2015
Number of different distributions of n+1 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Dec 31 2015
After the first three terms, partial sums of A008615. - Robert Israel, Dec 31 2015
For n >= 2, also the number of partitions of n - 2 into 3 parts. The Heinz numbers of these partitions are given by A014612. - Gus Wiseman, Oct 11 2020

			a(5) = a(6) = 1 with only one ordered triple (5,2,1). - _Michael Somos_, Feb 02 2015
a(11) = 5 Number of different distributions of 11 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
a(1) = a(2) = a(3) = a(4) = a(5) = 0, since with fewer than 6 identical balls there is no such distribution with 3 boxes that holds for 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
G.f.: x^5 + x^6 + 2*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 7*x^11 + 8*x^12 + ...
From _Gus Wiseman_, Oct 11 2020: (Start)
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three parts > 1 [Woodward] are the following (A = 10, B = 11, C = 12). The ordered version is A000217(n - 4) and the Heinz numbers are A046316.
  222  322  332  333  433  443  444  544  554  555  655
            422  432  442  533  543  553  644  654  664
                 522  532  542  552  643  653  663  754
                      622  632  633  652  662  744  763
                           722  642  733  743  753  772
                                732  742  752  762  844
                                822  832  833  843  853
                                     922  842  852  862
                                          932  933  943
                                          A22  942  952
                                               A32  A33
                                               B22  A42
                                                    B32
                                                    C22
The a(5) = 1 through a(15) = 14 partitions of n + 4 into three distinct parts > 1 [Resta] are the following (A = 10, B = 11, C = 12, D = 13, E = 14). The ordered version is A211540*6 and the Heinz numbers are A046389.
  432  532  542  543  643  653  654  754  764  765  865
            632  642  652  743  753  763  854  864  874
                 732  742  752  762  853  863  873  964
                      832  842  843  862  872  954  973
                           932  852  943  953  963  982
                                942  952  962  972  A54
                                A32  A42  A43  A53  A63
                                     B32  A52  A62  A72
                                          B42  B43  B53
                                          C32  B52  B62
                                               C42  C43
                                               D32  C52
                                                    D42
                                                    E32
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three distinct parts [Uslu and Becenen] are the following (A = 10, B = 11, C = 12, D = 13). The ordered version is A211540(n)*6 and the Heinz numbers are A007304.
  321  421  431  432  532  542  543  643  653  654  754
            521  531  541  632  642  652  743  753  763
                 621  631  641  651  742  752  762  853
                      721  731  732  751  761  843  862
                           821  741  832  842  852  871
                                831  841  851  861  943
                                921  931  932  942  952
                                     A21  941  951  961
                                          A31  A32  A42
                                          B21  A41  A51
                                               B31  B32
                                               C21  B41
                                                    C31
                                                    D21
(End)

Robert Israel, Table of n, a(n) for n = 0..10000
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Ece Uslu and Esin Becenen, Identical Object Distributions.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A001399, A069905, A211422.
All of the following pertain to 3-part strict partitions.
- A000009 counts these partitions of any length, with non-strict version A000041.
- A007304 gives the Heinz numbers, with non-strict version A014612.
- A101271 counts the relatively prime case, with non-strict version A023023.
- A220377 counts the pairwise coprime case, with non-strict version A307719.
- A337605 counts the pairwise non-coprime case, with non-strict version A337599.
Cf. A000217, A001840, A156040, A284825, A337453, A337483, A337484, A337563.

I:=[0,0,0,0,0,1]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015

f:= gfun:-rectoproc({a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6),seq(a(i)=0,i=0..4),a(5)=1},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, Dec 31 2015

t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 4 y, {w, n}, {x, n}, {y, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211540 *)
FindLinearRecurrence[t]
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 1}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[Length[Select[IntegerPartitions[n+1,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 05 2020 *)

{a(n) = round( (n-2)^2 / 12 )}; / * Michael Somos, Feb 02 2015 */

concat(vector(5), Vec(x^5/(1-x-x^2+x^4+x^5-x^6) + O(x^100))) \\ Altug Alkan, Jan 10 2016

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
a(n) = A069905(n-2) = A001399(n-5) for n >= 5. - Alois P. Heinz, Nov 03 2012
a(n) = 3*k^2-6*k+3 (for n = 6*k-3), 3*k^2-5*k+2 (for n = 6*k-2), 3*k^2-4*k+1 (for n = 6*k-1), 3*k^2-3*k+1 (for n = 6*k), 3*k^2-2*k (for n = 6*k+1), 3*k^2-k (for n = 6*k+2). - Ece Uslu, Esin Becenen, Dec 31 2015
a(n) = A004526(n-2) + a(n-2) for n > 2. - Ece Uslu, Esin Becenen, Dec 31 2015
G.f.: x^5/(1 - x - x^2 + x^4 + x^5 - x^6). - Robert Israel, Dec 31 2015
a(n) = Sum_{k=1..floor(n/3)} floor((n-k)/2)-k. - Wesley Ivan Hurt, Apr 27 2019
From Gus Wiseman, Oct 11 2020: (Start)
a(n+2) = A069905(n) = A001399(n-3) counts 3-part partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part strict partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part partitions with no 1's.
a(n-4) = A069905(n-6) = A001399(n-9) counts 3-part strict partitions with no 1's.
A000217(n-2) counts 3-part compositions.
a(n-1)*6 = A069905(n-3)*6 = A001399(n-6)*6 counts 3-part strict compositions.
A000217(n-5) counts 3-part compositions with no 1's.
a(n-4)*6 = A069905(n-6)*6 = A001399(n-9)*6 counts 3-part strict compositions with no 1's.
(End)

A130519 a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.)

Original entry on oeis.org

0, 0, 0, 0, 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

Offset: 0

Hieronymus Fischer, Jun 01 2007

Complementary to A130482 with respect to triangular numbers, in that A130482(n) + 4*a(n) = n(n+1)/2 = A000217(n).
Disregarding the first three 0's the resulting sequence a'(n) is the sum of the positive integers <= n that have the same residue modulo 4 as n. This is the additive counterpart of the quadruple factorial numbers. - Peter Luschny, Jul 06 2011
From Heinrich Ludwig, Dec 23 2017: (Start)
Column sums of (shift of rows = 4):
  1 2 3 4 5 6  7  8  9 10 11 12 13 14 ...
          1 2  3  4  5  6  7  8  9 10 ...
                     1  2  3  4  5  6 ...
                                 1  2 ...
  .......................................
  ---------------------------------------
  1 2 3 4 6 8 10 12 15 18 21 24 28 32 ...
  shift of rows = 1 see A000217
  shift of rows = 2 see A002620
  shift of rows = 3 see A001840
  shift of rows = 5 see A130520
(End)
Conjecture: a(n+2) is the maximum effective weight of a numerical semigroup S of genus n (see Nathan Pflueger). - Stefano Spezia, Jan 04 2019

			G.f. = x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + 10*x^10 + 12*x^11 + ...
[ n] a(n)
---------
[ 4] 1
[ 5] 2
[ 6] 3
[ 7] 4
[ 8] 1 + 5
[ 9] 2 + 6
[10] 3 + 7
[11] 4 + 8

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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.
Darren Glass and Joshua Wagner, Arithmetical Structures on Paths With a Doubled Edge, arXiv:1903.01398 [math.CO], 2019.
Nathan Pflueger, On non-primitive Weierstrass points, Alg. Number Th. 12 (2018) 1923-1947 and arXiv:1608.0566 [math.AG], 2016.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A000217, A001840, A002264, A002265, A002266, A002620, A004526, A010872, A010873, A010874, A130481, A130483, A130520.

Cf. A000290, A007590, A000212, A118015, A056827, A056834, A056838, A056865.

a:=List([0..65],n->Sum([0..n],k->Int(k/4)));; Print(a); # Muniru A Asiru, Jan 04 2019

[Round(n*(n-2)/8): n in [0..70]]; // Vincenzo Librandi, Jun 25 2011

quadsum := n -> add(k, k = select(k -> k mod 4 = n mod 4, [$1 .. n])):
A130519 := n ->`if`(n<3,0,quadsum(n-3)); seq(A130519(n),n=0..58); # Peter Luschny, Jul 06 2011

a[ n_] := Quotient[ (n - 1)^2, 8]; (* Michael Somos, Oct 14 2011 *)

makelist(floor((n-1)^2/8), n, 0, 70); /* Stefano Spezia, Jan 04 2019 */

{a(n) = (n - 1)^2 \ 8}; /* Michael Somos, Oct 14 2011 */

def A130519(n): return (n-1)**2>>3  # Chai Wah Wu, Jul 30 2022

G.f.: x^4/((1-x^4)*(1-x)^2) = x^4/((1+x)*(1+x^2)*(1-x)^3).
a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-4) -2*a(n-5) +1*a(n-6).
a(n) = floor(n/4)*(n - 1 - 2*floor(n/4)) = A002265(n)*(n - 1 - 2*A002265(n)).
a(n) = (1/2)*A002265(n)*(n - 2 + A010873(n)).
a(n) = floor((n-1)^2/8). - Mitch Harris, Sep 08 2008
a(n) = round(n*(n-2)/8) = round((n^2-2*n-1)/8) = ceiling((n+1)*(n-3)/8). - Mircea Merca, Nov 28 2010
a(n) = A001972(n-4), n>3. - Franklin T. Adams-Watters, Jul 10 2009
a(n) = a(n-4)+n-3, n>3. - Mircea Merca, Nov 28 2010
Euler transform of length 4 sequence [ 2, 0, 0, 1]. - Michael Somos, Oct 14 2011
a(n) = a(2-n) for all n in Z. - Michael Somos, Oct 14 2011
a(n) = A214734(n, 1, 4). - Renzo Benedetti, Aug 27 2012
a(4n) = A000384(n), a(4n+1) = A001105(n), a(4n+2) = A014105(n), a(4n+3) = A046092(n). - Philippe Deléham, Mar 26 2013
a(n) = Sum_{i=1..ceiling(n/2)-1} (i mod 2) * (n - 2*i - 1). - Wesley Ivan Hurt, Jan 23 2014
a(n) = ( 2*n^2-4*n-1+(-1)^n+2*((-1)^((2*n-1+(-1)^n)/4)-(-1)^((6*n-1+(-1)^n)/4)) )/16 = ( 2*n*(n-2) - (1-(-1)^n)*(1-2*i^(n*(n-1))) )/16, where i=sqrt(-1). - Luce ETIENNE, Aug 29 2014
E.g.f.: (1/8)*((- 1 + x)*x*cosh(x) + 2*sin(x) + (- 1 - x + x^2)*sinh(x)). - Stefano Spezia, Jan 15 2019
a(n) = (A002620(n-1) - A011765(n+1)) / 2, for n > 0. - Yuchun Ji, Feb 05 2021
Sum_{n>=4} 1/a(n) = Pi^2/12 + 5/2. - Amiram Eldar, Aug 13 2022

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

A014125 Bisection of A001400.

Original entry on oeis.org

1, 3, 6, 11, 18, 27, 39, 54, 72, 94, 120, 150, 185, 225, 270, 321, 378, 441, 511, 588, 672, 764, 864, 972, 1089, 1215, 1350, 1495, 1650, 1815, 1991, 2178, 2376, 2586, 2808, 3042, 3289, 3549, 3822, 4109, 4410, 4725, 5055, 5400, 5760, 6136, 6528, 6936, 7361, 7803

Offset: 0

N. J. A. Sloane

Also Schoenheim bound L_1(n,5,4).
Degrees of polynomials defined by p(n) = (x^(n+1)*p(n-1)p(n-3) + p(n-2)^2)/p(n-4), p(-4)=p(-3)=p(-2)=p(-1)=1. - Michael Somos, Jul 21 2004
Degrees of polynomial tau-functions of q-discrete Painlevé I, which generate sequence A095708 when q=2 (up to an offset of 3). - Andrew Hone, Jul 29 2004
Because of the Laurent phenomenon for the general q-discrete Painlevé I tau-function recurrence p(n) = (a*x^(n+1)*p(n-1)*p(n-3) + b*p(n-2)^2)/p(n-4), p(n) for n > -1 will always be a polynomial in x and a Laurent polynomial in a,b and the initial data p(-4),p(-3),p(-2),p(-1). - Andrew Hone, Jul 29 2004
Create the sequence 0,0,0,0,0,6,18,36,66,108,... so that the sum of three consecutive terms b(n) + b(n+1) + b(n+2) = A007531(n), with b(0)=0; then a(n) = b(n+5)/6. - J. M. Bergot, Jul 30 2013
Number of partitions of n into three kinds of part 1 and one kind of part 3. - Joerg Arndt, Sep 28 2015
First differences are A001840(k) starting with k=2; second differences are A086161(k) starting with k=1. - Bob Selcoe, Sep 28 2015
Maximum Wiener index of all maximal planar graphs with n+2 vertices.  The extremal graphs are cubes of paths. - Allan Bickle, Jul 09 2022
Maximum Wiener index of all maximal 3-degenerate graphs with n+2 vertices.  (A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices.)  The extremal graphs are cubes of paths, so the bound also applies to 3-trees. - Allan Bickle, Sep 18 2022

			Polynomials: p(0)=x+1, p(1)=x^3+x^2+1, p(2)=x^6+x^5+x^3+x^2+2x+1, ...
a(12)=185:  A000217(13)=91 + a(9)=94 == 91+55+28+10+1 = 185. - _Bob Selcoe_, Sep 27 2015
a(3)=11: the 11 partitions of 3 are {1a,1a,1a}, {1a,1a,1b}, {1a,1a,1c}, {1a,1b,1b}, {1a,1b,1c}, {1a,1c,1c}, {1b,1b,1b}, {1b,1b,1c}, {1b,1c,1c}, {1c,1c,1c}, {3}. - _Bob Selcoe_, Oct 04 2015

W. H. Mills and R. C. Mullin, Coverings and packings, pp. 371-399 of Jeffrey H. Dinitz and D. R. Stinson, editors, Contemporary Design Theory, Wiley, 1992. See Eq. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
L. Smiley, Hidden Hexagons (preprint).

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Z. Che and K. L. Collins, An upper bound on Wiener indices of maximal planar graphs, Discrete Appl. Math. 258 (2019), 76-86.
Éva Czabarka, Peter Dankelmann, Trevor Olsen, and László A. Székely, Wiener Index and Remoteness in Triangulations and Quadrangulations, arXiv:1905.06753 [math.CO], 2019.
S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics, 28 (2002), 119-144.
D. Ghosh, E. Győri, A. Paulos, N. Salia, and O. Zamora, The maximum Wiener index of maximal planar graphs, Journal of Combinatorial Optimization 40, (2020), 1121-1135.
H. R. Henze and C. M. Blair, The number of structurally isomeric hydrocarbons of the ethylene series, J. Amer. Chem. Soc., 55 (1933), 680-685.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, arXiv:0807.2538 [nlin.SI], 2008; Proceedings of SIDE 6, Helsinki, Finland, 2004. [Set a(n)=d(n+3) on p. 8]
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=3]
Index entries for two-way infinite sequences
Index entries for covering numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A000217, A000631, A001840, A014126, A086161, A095708.
A column of A036838.
Cf. A002623, A014125, A122046, A122047. - Johannes W. Meijer, May 20 2011
Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046, A122047, A175724.

[n^3/18+n^2/2+4*n/3+1+(((n+1) mod 3)-1)/9 : n in [0..50]]; // Wesley Ivan Hurt, Apr 14 2015

I:=[1,3,6,11,18,27]; [n le 6 select I[n] else 3*Self(n-1) -3*Self(n-2) +2*Self(n-3)-3*Self(n-4)+3*Self(n-5)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Apr 15 2015

L := proc(v,k,t,l) local i,t1; t1 := l; for i from v-t+1 to v do t1 := ceil(t1*i/(i-(v-k))); od: t1; end; # gives Schoenheim bound L_l(v,k,t). Current sequence is L_1(n,n-3,n-4,1).

CoefficientList[Series[1/((1 - x)^3*(1 - x^3)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Apr 14 2015 *)

a(n)=if(n<-5,-a(-6-n),polcoeff(1/(1-x)^3/(1-x^3)+x^n*O(x),n)) /* Michael Somos, Jul 21 2004 */

my(x='x+O('x^50)); Vec(1/((1-x)^3*(1-x^3))) \\ Altug Alkan, Oct 16 2015

a(n)=(n^3 + 9*n^2 + 24*n + 19)\/18 \\ Charles R Greathouse IV, Jun 29 2020

[(binomial(n+4,3) - ((n+4)//3))/3 for n in (0..50)] # G. C. Greubel, Apr 28 2019

G.f.: 1/((1-x)^3*(1-x^3)).
a(n) = -a(-6-n) = 3*a(n-1) -3*a(n-2) +2*a(n-3) -3*a(n-4) +3*a(n-5) -a(n-6).
The simplest recurrence is fourth order: a(n) = a(n-1) + a(n-3) - a(n-4) + n + 1, which gives the g.f.: 1/((1-x)^3*(1-x^3)), with a(-4) = a(-3) = a(-2) = a(-1) = 0.
a(n) = n^3/18 + n^2/2 + 4*n/3 + 1 + 2/(9*sqrt(3))*sin(2*Pi*n/3). - Andrew Hone, Jul 29 2004
a(n) = n^3/18 + n^2/2 + 4*n/3 + 1 + (((n+1) mod 3) - 1)/9. - same formula, simplified by Gerald Hillier, Apr 14 2015
a(n) = (2*A000027(n+1) + 3*A000292(n+1) + A049347(n-1) + 1 + 3*A000217(n+1))/9. - R. J. Mathar, Nov 16 2007
From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A144677(n) + A144677(n-1) + A144677(n-2).
a(n) = A190717(n-4) + 2*A190717(n-3) + 3*A190717(n-2) + 2*A190717(n-1) + A190717(n). (End)
3*a(n) = binomial(n+4,3) - floor((n+4)/3). - Bruno Berselli, Nov 08 2013
a(n) = A000217(n+1) + a(n-3) = Sum_{j>=0, n>=3*j} (n-3*j+1)*(n-3*j+2)/2. - Bob Selcoe, Sep 27 2015
a(n) = round(((2*n+5)^3 + 3*(2*n+5)^2 - 9*(2*n+5))/144). - Giacomo Guglieri, Jun 28 2020
a(n) = floor(((n+2)^3 + 3*(n+2)^2)/18). - Allan Bickle, Aug 01 2020
a(n) = Sum_{j=0..n} (n-j+1)*floor((j+3)/3). - G. C. Greubel, Oct 18 2021
E.g.f.: exp(x) + exp(x)*x*(34 + 12*x + x^2)/18 + 2*exp(-x/2)*sin(sqrt(3)*x/2)/(9*sqrt(3)). - Stefano Spezia, Apr 05 2023

More terms from James Sellers, Dec 24 1999

A130518 a(n) = Sum_{k=0..n} floor(k/3). (Partial sums of A002264.)

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570

Offset: 0

Hieronymus Fischer, Jun 01 2007

Complementary with A130481 regarding triangular numbers, in that A130481(n) + 3*a(n) = n(n+1)/2 = A000217(n).
Apart from offset, the same as A062781. - R. J. Mathar, Jun 13 2008
Apart from offset, the same as A001840. - Michael Somos, Sep 18 2010
The sum of any three consecutive terms is a triangular number. - J. M. Bergot, Nov 27 2014

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

Cf. A002265, A002266, A004526, A010872, A010873, A010874, A062781, A130482, A130483.

List([0..60], n-> Int(n*(n-1)/6)); # G. C. Greubel, Aug 31 2019

[Round(n*(n-1)/6): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

seq(floor(n*(n-1)/6),n=0..60); # Robert Israel, Nov 27 2014

Table[n, {n, 0, 19}, {3}] // Flatten // Accumulate (* Jean-François Alcover, Jun 05 2013 *)

a(n)=n*(n-1)\/6 \\ Charles R Greathouse IV, Jun 05 2013

[floor(binomial(n,2)/3) for n in range(0,60)] # Zerinvary Lajos, Dec 01 2009

G.f.: x^3 / ((1-x^3)*(1-x)^2).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(n) = (1/2)*floor(n/3)*(2*n - 1 - 3*floor(n/3)) = A002264(n)*(2n - 1 - 3*A002264(n))/2.
a(n) = (1/2)*A002264(n)*(n - 1 + A010872(n)).
a(n) = round(n*(n-1)/6) = round((n^2-n-1)/6) = floor(n*(n-1)/6) = ceiling((n+1)*(n-2)/6). - Mircea Merca, Nov 28 2010
a(n) = a(n-3) + n - 2, n > 2. - Mircea Merca, Nov 28 2010
a(n) = A214734(n, 1, 3). - Renzo Benedetti, Aug 27 2012
a(3n) = A000326(n), a(3n+1) = A005449(n), a(3n+2) = 3*A000217(n) = A045943(n). - Philippe Deléham, Mar 26 2013
a(n) = (3*n*(n-1) - (-1)^n*((1+i*sqrt(3))^(n-2) + (1-i*sqrt(3))^(n-2))/2^(n-3) - 2)/18, where i=sqrt(-1). - Bruno Berselli, Nov 30 2014
Sum_{n>=3} 1/a(n) = 20/3 - 2*Pi/sqrt(3). - Amiram Eldar, Sep 17 2022

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A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

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

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A051168 Triangular array T(h,k) for 0 <= k <= h read by rows: T(h,k) = number of binary Lyndon words with k ones and h-k zeros.

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A008620 Positive integers repeated three times.

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A337461 Number of pairwise coprime ordered triples of positive integers summing to n.

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A007997 a(n) = ceiling((n-3)(n-4)/6).

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