A053618 -id:A053618 - cp's OEIS frontend

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

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

A032192 Number of necklaces with 7 black beads and n-7 white beads.

Original entry on oeis.org

1, 1, 4, 12, 30, 66, 132, 246, 429, 715, 1144, 1768, 2652, 3876, 5538, 7752, 10659, 14421, 19228, 25300, 32890, 42288, 53820, 67860, 84825, 105183, 129456, 158224, 192130, 231880, 278256, 332112, 394383, 466089, 548340

Offset: 7

Christian G. Bower

nonn

"CIK[ 7 ]" (necklace, indistinct, unlabeled, 7 parts) transform of 1, 1, 1, 1, ...
The g.f. is Z(C_7,x)/x^7, the 7-variate cycle index polynomial for the cyclic group C_7, with substitution x[i]->1/(1-x^i), i=1,...,7. Therefore by Polya enumeration a(n+7) is the number of cyclically inequivalent 7-necklaces whose 7 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_7,x) and the comment in A032191 on the equivalence of this problem with the one given in the 'Name' line. - Wolfdieter Lang, Feb 15 2005
From Petros Hadjicostas, Dec 08 2017: (Start)
For p prime, if a_p(n) is the number of necklaces with p black beads and n-p white beads, then (a_p(n): n>=1) = CIK[p](1, 1, 1, 1, ...). Since CIK[k](B(x)) = (1/k)*Sum_{d|k} phi(d)*B(x^d)^{k/d} with k = p and B(x) = x + x^2 + x^3 + ... = x/(1-x), we get Sum_{n>=1} a_p(n)*x^n = ((p-1)/(1 - x^p) + 1/(1 - x)^p)*x^p/p, which is Herbert Kociemba's general formula for the g.f. when p is prime.
We immediately get a_p(n) = ((p-1)/p)*I(p|n) + (1/p)*C(n-1,p-1) = ((p-1)/p)*I(p|n) + (1/n)*C(n,p) = ceiling(C(n,p)/n), which is a generalization of the conjecture made by N. J. A. Sloane and Wolfdieter Lang. (Here, I(condition) = 1 if the condition holds, and 0 otherwise. Also, as usual, for integers n and k, C(n,k) = 0 if 0 <= n < k.)
Since the sequence (a_p(n): n>=1) is column k = p of A047996(n,k) = T(n,k), we get from the documentation of the latter sequence that a_p(n) = T(n, p) = (1/n)*Sum_{d|gcd(n,p)} phi(d)*C(n/d, p/d), from which we get another proof of the formulae for a_p(n).
(End)

C. G. Bower, Transforms (2)
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
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 pp. 5-6.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1,1,-6,15,-20,15,-6,1).

Column k=7 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A053618.

k = 7; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
DeleteCases[CoefficientList[Series[x^7 (x^6 - 5 x^5 + 13 x^4 - 17 x^3 + 13 x^2 - 5 x + 1)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (1 - x)^7), {x, 0, 41}], x], 0] (* Michael De Vlieger, Oct 10 2016 *)

Empirically this is ceiling(C(n, 7)/n). - N. J. A. Sloane
G.f.: x^7*(x^6 - 5*x^5 + 13*x^4 - 17*x^3 + 13*x^2 - 5*x + 1)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(1 - x)^7). - Gael Linder (linder.gael(AT)wanadoo.fr), Jan 13 2005
G.f.: (6/(1 - x^7) + 1/(1 - x)^7)*x^7/7; in general, for a necklace with p black beads and p prime, the g.f. is ((p-1)/(1 - x^p) + 1/(1 - x)^p)*x^p/p. - Herbert Kociemba, Oct 15 2016
a(n) = ceiling(binomial(n, 7)/n) (conjecture by Wolfdieter Lang).
a(n) = (6/7)*I(7|n) + (1/7)*C(n-1,6) = (6/7)*I(7|n) + (1/n)*C(n,7), where I(condition) = 1 if the condition holds, and = 0 otherwise. - Petros Hadjicostas, Dec 08 2017

A053643 a(n) = ceiling(binomial(n,6)/n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 4, 10, 21, 42, 77, 132, 215, 334, 501, 728, 1032, 1428, 1938, 2584, 3392, 4389, 5609, 7084, 8855, 10964, 13455, 16380, 19793, 23751, 28319, 33563, 39556, 46376, 54106, 62832, 72650, 83657, 95960, 109668, 124900

Offset: 1

N. J. A. Sloane, Mar 25 2000

Robert Israel, Table of n, a(n) for n = 1..10000
R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1,0,0,0,0,0,0,0,0,0,0,0,0,1,-5,10,-10,5,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,5,-10,10,-5,1,0,0,0,0,0,0,0,0,0,0,0,0,1,-5,10,-10,5,-1).

Cf. A000389, A000579, A053618.

[Ceiling(Binomial(n,6)/n) : n in [1..50]]; // Wesley Ivan Hurt, Nov 01 2015

seq(ceil(binomial(n,5)/6), n=0..50); # Zerinvary Lajos, Jan 12 2009

Table[Ceiling[Binomial[n, 6]/n], {n, 50}] (* Michael De Vlieger, Nov 01 2015 *)

vector(50, n, ceil(binomial(n, 6)/n)) \\ Altug Alkan, Nov 01 2015

[ceil(binomial(n,6)/n) for n in (1..50)] # G. C. Greubel, May 17 2019

From Robert Israel, Nov 01 2015: (Start)
a(n) = ceiling(A000389(n-1)/6).
G.f.: (x^52 -2*x^51 +4*x^50 -4*x^49 +2*x^48 +x^47 -x^46 +2*x^44 -2*x^43 +x^42 -x^41 +4*x^40 -5*x^39 +4*x^38 -2*x^37 +3*x^36 -5*x^35 +5*x^34 -2*x^33 -2*x^32 +5*x^31 -5*x^30 +2*x^29 +2*x^28 -5*x^27 +5*x^26 -2*x^25 -2*x^24 +5*x^23 -5*x^22 +2*x^21 +2*x^20 -5*x^19 +5*x^18 -3*x^17 +4*x^16 -7*x^15 +9*x^14 -7*x^13 +5*x^12 -4*x^11 +4*x^10 -2*x^9 +3*x^7 -4*x^6 +x^5 +6*x^4 -10*x^3 +9*x^2 -4*x +1)*x^6/((x -1)^6*(x +1)*(x^4 +1)*(x^2 +x +1)*(x^2 -x +1)*(x^6 +x^3 +1)*(x^6 -x^3 +1)*(x^8 -x^4 +1)*(x^24 -x^12 +1)).
(End)

A053731 a(n) = ceiling(binomial(n,8)/n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 15, 42, 99, 215, 429, 805, 1430, 2431, 3978, 6299, 9690, 14535, 21318, 30645, 43263, 60088, 82225, 111004, 148005, 195098, 254475, 328697, 420732, 534006, 672452, 840565, 1043460, 1286934, 1577532, 1922618

Offset: 1

N. J. A. Sloane, Mar 25 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-7,21,-35,35,-21,7,-1).

Cf. Sequences of the form ceiling(binomial(n,k)/n): A000012 (k=1), A004526 (k=2), A007997 (k=3), A008646 (k=5), A032192 (k=7), A053618 (k=4), A053643 (k=6), this sequence (k=8), A053733 (k=9).

[Ceiling(Binomial(n,8)/n): n in [1..45]]; // G. C. Greubel, Sep 06 2019

seq(ceil(binomial(n,8)/n), n=1..45); # G. C. Greubel, Sep 06 2019

Table[Ceiling[Binomial[n, 8]/n], {n, 45}] (* G. C. Greubel, Sep 06 2019 *)

vector(45, n, ceil(binomial(n,8)/n)) \\ G. C. Greubel, Sep 06 2019

[ceil(binomial(n,8)/n) for n in (1..45)] # G. C. Greubel, Sep 06 2019

A053733 a(n) = ceiling(binomial(n,9)/n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 19, 55, 143, 334, 715, 1430, 2702, 4862, 8398, 13997, 22610, 35530, 54480, 81719, 120175, 173587, 246675, 345345, 476905, 650325, 876525, 1168700, 1542684, 2017356, 2615092, 3362260, 4289780, 5433722

Offset: 1

N. J. A. Sloane, Mar 25 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-8,28,-56,70,-56,28,-8,1).

Cf. Sequences of the form ceiling(binomial(n,k)/n): A000012 (k=1), A004526 (k=2), A007997 (k=3), A008646 (k=5), A032192 (k=7), A053618 (k=4), A053643 (k=6), A053731 (k=8), this sequence (k=9).

[Ceiling(Binomial(n,9)/n): n in [1..40]]; // G. C. Greubel, Sep 06 2019

seq(ceil(binomial(n,9)/n), n=1..40); # G. C. Greubel, Sep 06 2019

Table[Ceiling[Binomial[n, 9]/n], {n, 40}] (* G. C. Greubel, Sep 06 2019 *)

vector(40, n, ceil(binomial(n,9)/n)) \\ G. C. Greubel, Sep 06 2019

[ceil(binomial(n,9)/n) for n in (1..40)] # G. C. Greubel, Sep 06 2019

Showing 1-5 of 5 results.

cp's OEIS Frontend

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

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A032192 Number of necklaces with 7 black beads and n-7 white beads.

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A053643 a(n) = ceiling(binomial(n,6)/n).

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A053731 a(n) = ceiling(binomial(n,8)/n).

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A053733 a(n) = ceiling(binomial(n,9)/n).

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