A011847 -id:A011847 - cp's OEIS frontend

A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).

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, 590

Offset: 0

N. J. A. Sloane

a(n-3) is the number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.
Number of triangular partitions (see Almkvist).
Consists of arithmetic progression quadruples of common difference n+1 starting at A045943(n). Refers to the least number of coins needed to be rearranged in order to invert the pattern of a (n+1)-rowed triangular array. For instance, a 5-rowed triangular array requires a minimum of a(4)=5 rearrangements (shown bracketed here) for it to be turned upside down.
  .....{*}..................{*}*.*{*}{*}
  .....*.*....................*.*.*.{*}
  ....*.*.*....---------\......*.*.*
  ..{*}*.*.*...---------/.......*.*
  {*}{*}*.*{*}..................{*}
  - Lekraj Beedassy, Oct 13 2003
Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry, Mar 01 2004
Sum of three successive terms is a triangular number in natural order starting with 3: a(n)+a(n+1)+a(n+2) = T(n+2) = (n+2)*(n+3)/2. - Amarnath Murthy, Apr 25 2004
Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry, Apr 16 2005
Absolute values of numbers that appear in A145919. - Matthew Vandermast, Oct 28 2008
In the Moree definition, (-1)^n*a(n) is the 3rd Witt transform of A033999 and (-1)^n*A004524(n) with 2 leading zeros dropped is the 2nd Witt transform of A033999. - R. J. Mathar, Nov 08 2008
Column sums of:
1 2 3 4 5 6 7  8  9.....
      1 2 3 4  5  6.....
            1  2  3.....
........................
----------------------
1 2 3 5 7 9 12 15 18  - Jon Perry, Nov 16 2010
a(n) is the sum of the positive integers <= n that have the same residue modulo 3 as n. They are the additive counterpart of the triple factorial numbers. - Peter Luschny, Jul 06 2011
a(n+1) is the number of 3-tuples (w,x,y) with all terms in {0,...,n} and w=3*x+y. - Clark Kimberling, Jun 04 2012
a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x-y = (1 mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012
a(n+1) is the number of partitions of n into two sorts of part(s) 1 and one sort of (part) 3. - Joerg Arndt, Jun 10 2013
Arrange A004523 in rows successively shifted to the right two spaces and sum the columns:
1  2  2  3  4  4  5  6  6...
      1  2  2  3  4  4  5...
            1  2  2  3  4...
                  1  2  2...
                        1...
------------------------------
1  2  3  5  7  9 12 15 18... - L. Edson Jeffery, Jul 30 2014
a(n) = A258708(n+1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015
Also the number of triples of positive integers summing to n + 4, the first less than each of the other two. Also the number of triples of positive integers summing to n + 2, the first less than or equal to each of the other two. - Gus Wiseman, Oct 11 2020
Also the lower matching number of the (n+1)-triangular honeycomb king graph = n-triangular grid graph (West convention). - Eric W. Weisstein, Dec 14 2024

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 12*x^7 + 15*x^8 + 18*x^9 + ...
1+2+3=6=t(3), 2+3+5=t(4), 5+7+9=t(5).
[n] a(n)
--------
[1] 1
[2] 2
[3] 3
[4] 1 + 4
[5] 2 + 5
[6] 3 + 6
[7] 1 + 4 + 7
[8] 2 + 5 + 8
[9] 3 + 6 + 9
a(7) = floor(2/3) +floor(3/3) +floor(4/3) +floor(5/3) +floor(6/3) +floor(7/3) +floor(8/3) +floor(9/3) = 12. - _Bruno Berselli_, Aug 29 2013

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
Ulrich Faigle, Review of Gerhard Post and G.J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, MR2224983(2007b:90134), 2007.
Hansraj Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
Richard K. Guy, A problem of Zarankiewicz, in P. Erdős and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150, (p. 126, divided by 2).
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).

T. D. Noe, Table of n, a(n) for n = 0..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.
Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Cristian Cobeli, Aaditya Raghavan, and Alexandru Zaharescu, On the central ball in a translation invariant involutive field, arXiv:2408.01864 [math.NT], 2024. See p. 7.
Neville de Mestre and John Baker, Pebbles, Ducks and Other Surprises, Australian Maths. Teacher, Vol. 48, No 3, 1992, pp. 4-7.
Peter M. Chema, Illustration of first 27 terms as corners of a double hexagon spiral from 0
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
Richard K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 207.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=3]
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.
Gerhard Post and G. J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, Discrete Optimization 3, pp. 165-173, 2006.
Michael Somos, Somos Polynomials.
Gary E. Stevens, A Connell-Like Sequence, J. Integer Seqs., 1 (1998), Article 98.1.4.
Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 7.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb King Graph.
Index entries for sequences related to Lyndon words.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000031, A001037, A008748, A051168.
Ordered union of triangular matchstick numbers A045943 and generalized pentagonal numbers A001318.
Cf. A058937.
A column of triangle A011847.
Cf. A000217, A130205.
Cf. A258708.
A001399 counts 3-part partitions, ranked by A014612.
A337483 counts either weakly increasing or weakly decreasing triples.
A337484 counts neither strictly increasing nor strictly decreasing triples.
A014311 ranks 3-part compositions, with strict case A337453.
Cf. A007052, A007304, A011782, A069905, A156040, A218004.

a001840 n = a001840_list !! n
a001840_list = scanl (+) 0 a008620_list
-- Reinhard Zumkeller, Apr 16 2012

[ n le 2 select n else n*(n+1)/2-Self(n-1)-Self(n-2): n in [1..58] ];  // Klaus Brockhaus, Oct 01 2009

A001840 := n->floor((n+1)*(n+2)/6);
A001840:=-1/((z**2+z+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
seq(floor(binomial(n-1,2)/3), n=3..61); # Zerinvary Lajos, Jan 12 2009
A001840 :=  n -> add(k, k = select(k -> k mod 3 = n mod 3, [$1 .. n])): seq(A001840(n), n = 0 .. 58); # Peter Luschny, Jul 06 2011

a[0]=0; a[1]=1; a[n_]:= a[n]= n(n+1)/2 -a[n-1] -a[n-2]; Table[a[n], {n,0,100}]
f[n_] := Floor[(n + 1)(n + 2)/6]; Array[f, 59, 0] (* Or *)
CoefficientList[ Series[ x/((1 + x + x^2)*(1 - x)^3), {x, 0, 58}], x] (* Robert G. Wilson v *)
a[ n_] := With[{m = If[ n < 0, -3 - n, n]}, SeriesCoefficient[ x /((1 - x^3) (1 - x)^2), {x, 0, m}]]; (* Michael Somos, Jul 11 2011 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,5},60] (* Harvey P. Dale, Jul 25 2011 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+4,{3}],#[[1]]<#[[2]]&&#[[1]]<#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020 *)

{a(n) = (n+1) * (n+2) \ 6}; /* Michael Somos, Feb 11 2004 */

[binomial(n, 2) // 3 for n in range(2, 61)] # Zerinvary Lajos, Dec 01 2009

a(n) = (A000217(n+1) - A022003(n-1))/3;
a(n) = (A016754(n+1) - A010881(A016754(n+1)))/24;
a(n) = (A033996(n+1) - A010881(A033996(n+1)))/24.
Euler transform of length 3 sequence [2, 0, 1].
a(3*k-1) = k*(3*k + 1)/2;
a(3*k)   = 3*k*(k + 1)/2;
a(3*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = floor( (n+1)*(n+2)/6 ) = floor( A000217(n+1)/3 ).
a(n+1) = a(n) + A008620(n) = A002264(n+3). - Reinhard Zumkeller, Aug 01 2002
From Michael Somos, Feb 11 2004: (Start)
G.f.: x / ((1-x)^2 * (1-x^3)).
a(n) = 1 + a(n-1) + a(n-3) - a(n-4).
a(-3-n) = a(n). (End)
a(n) = a(n-3) + n for n > 2; a(0)=0, a(1)=1, a(2)=2. - Paul Barry, Jul 14 2004
a(n) = binomial(n+3, 3)/(n+3) + cos(2*Pi*(n-1)/3)/9 + sqrt(3)sin(2*Pi*(n-1)/3)/9 - 1/9. - Paul Barry, Jan 01 2005
From Paul Barry, Apr 16 2005: (Start)
a(n) = Sum_{k=0..n} k*(cos(2*Pi*(n-k)/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3).
a(n) = Sum_{k=0..floor(n/3)} n-3*k. (End)
For n > 1, a(n) = A000217(n) - a(n-1) - a(n-2); a(0)=0, a(1)=1.
G.f.: x/(1 + x + x^2)/(1 - x)^3. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = (4 + 3*n^2 + 9*n)/18 + ((n mod 3) - ((n-1) mod 3))/9. - Klaus Brockhaus, Oct 01 2009
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), with n>4, a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - Harvey P. Dale, Jul 25 2011
a(n) = A214734(n + 2, 1, 3). - Renzo Benedetti, Aug 27 2012
G.f.: x*G(0), where G(k) = 1 + x*(3*k+4)/(3*k + 2 - 3*x*(k+2)*(3*k+2)/(3*(1+x)*k + 6*x + 4 - x*(3*k+4)*(3*k+5)/(x*(3*k+5) + 3*(k+1)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 10 2013
Empirical: a(n) = floor((n+3)/(e^(6/(n+3))-1)). - Richard R. Forberg, Jul 24 2013
a(n) = Sum_{i=0..n} floor((i+2)/3). - Bruno Berselli, Aug 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) = n/2 + floor(n^2/3 + 2/3)/2. - Bruno Berselli, Jan 23 2017
a(n) + a(n+1) = A000212(n+2). - R. J. Mathar, Jan 14 2021
Sum_{n>=1} 1/a(n) = 20/3 - 2*Pi/sqrt(3). - Amiram Eldar, Sep 27 2022
E.g.f.: (exp(x)*(4 + 12*x + 3*x^2) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/18. - Stefano Spezia, Apr 05 2023

A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 3, 3, 0, 0, 4, 4, 4, 0, 0, 5, 10, 10, 5, 0, 0, 6, 12, 18, 12, 6, 0, 0, 7, 21, 35, 35, 21, 7, 0, 0, 8, 24, 56, 64, 56, 24, 8, 0, 0, 9, 36, 81, 126, 126, 81, 36, 9, 0, 0, 10, 40, 120, 200, 250, 200, 120, 40, 10, 0, 0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11

Offset: 0

Clark Kimberling

			For example, T(4,2) counts 1100,1001,0011,0110; T(2,1) counts 10, 01 (hence also counts 1010, 0101).
Rows:
  1;
  1,  1;
  0,  2,  0;
  0,  3,  3,  0;
  0,  4,  4,  4,  0;
  0,  5, 10, 10,  5,  0;

M. F. Hasler, A050186, rows 0..50, Sep 27 2018
M. F. Hasler, A050186, rows 0..100, Sep 27 2018
M. F. Hasler, A050186, rows 0..200, Sep 27 2018
N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Same triangle as A053727 except this one includes column 0.

T(2n, n), T(2n+1, n) match A007727, A001700, respectively. Row sums match A027375.

T[n_, k_] := If[n == 0, 1, DivisorSum[GCD[k, n], MoebiusMu[#] Binomial[n/#, k/#]&]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 16 2022 *)

A050186(n,k)=sumdiv(gcd(n+!n,k),d,moebius(d)*binomial(n/d,k/d)) \\ M. F. Hasler, Sep 27 2018

MOEBIUS transform of A007318 Pascal's Triangle.

If rows n > 1 are divided by n, this yields the triangle A051168, which equals A245558 surrounded by 0's (except for initial terms). This differs from A011847 from row n = 9 on. - M. F. Hasler, Sep 29 2018

A011857 Triangle of numbers [ C(n,k)/k ], k=1..n-1.

Original entry on oeis.org

2, 3, 1, 4, 3, 1, 5, 5, 3, 1, 6, 7, 6, 3, 1, 7, 10, 11, 8, 4, 1, 8, 14, 18, 17, 11, 4, 1, 9, 18, 28, 31, 25, 14, 5, 1, 10, 22, 40, 52, 50, 35, 17, 5, 1, 11, 27, 55, 82, 92, 77, 47, 20, 6, 1, 12, 33, 73, 123, 158, 154, 113, 61, 24, 6, 1, 13, 39, 95, 178, 257, 286, 245, 160

Offset: 2

N. J. A. Sloane

Columns include A011848, A011849, A011850, A011851, A011852, A011853, A011854, A011855, A011856. Row sums are in A101687. Cf. A011847.

Flatten[Table[Floor[Binomial[n,k]/k],{n,20},{k,n-1}]] (* Harvey P. Dale, Apr 19 2015 *)

A245558 Square array read by antidiagonals: T(n,k) = number of n-tuples of nonnegative integers (u_0,...,u_{n-1}) satisfying Sum_{j=0..n-1} j*u_j == 1 mod n and Sum_{j=0..n-1} u_j = m.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 3, 7, 8, 7, 3, 1, 1, 4, 9, 14, 14, 9, 4, 1, 1, 4, 12, 20, 25, 20, 12, 4, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 5, 18, 40, 66, 75, 66, 40, 18, 5, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1

Offset: 1

N. J. A. Sloane, Aug 07 2014

The array is symmetric; for the entries on or below the diagonal see A245559.
If the congruence in the definition is changed from Sum_{j=0..n-1} j*u_j == 1 mod n  to Sum_{j=0..n-1} j*u_j == 0 mod n we get the array shown in A241926, A047996, and A037306.
Differs from A011847 from row n = 9, k = 4 on; if the rows are surrounded by 0's, this yields A051168 without its rows 0 and 1, i.e., a(1) is A051168(2,1). - M. F. Hasler, Sep 29 2018
This array was first studied by Fredman (1975). - Petros Hadjicostas, Jul 10 2019

			Square array begins:
  1, 1,  1,  1,   1,   1,    1,    1,    1,    1, ...
  1, 1,  2,  2,   3,   3,    4,    4,    5,    5, ...
  1, 2,  3,  5,   7,   9,   12,   15,   18,   22, ...
  1, 2,  5,  8,  14,  20,   30,   40,   55,   70, ...
  1, 3,  7, 14,  25,  42,   66,   99,  143,  200, ...
  1, 3,  9, 20,  42,  75,  132,  212,  333,  497, ...
  1, 4, 12, 30,  66, 132,  245,  429,  715, 1144, ...
  1, 4, 15, 40,  99, 212,  429,  800, 1430, 2424, ...
  1, 5, 18, 55, 143, 333,  715, 1430, 2700, 4862, ...
  1, 5, 22, 70, 200, 497, 1144, 2424, 4862, 9225, ...
  ...
Reading by antidiagonals, we get:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 2,  3,  2,  1;
  1, 3,  5,  5,  3,   1;
  1, 3,  7,  8,  7,   3,   1;
  1, 4,  9, 14, 14,   9,   4,  1;
  1, 4, 12, 20, 25,  20,  12,  4,  1;
  1, 5, 15, 30, 42,  42,  30, 15,  5,  1;
  1, 5, 18, 40, 66,  75,  66, 40, 18,  5, 1;
  1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1;
  ...

Taylor Brysiewicz, Necklaces count polynomial parametric osculants, arXiv:1807.03408 [math.AG], 2018.
A. Elashvili, 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, D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173-188. MR1719140 (2000j:05009). See p. 174.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
I. M. Gessel and C. Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory, Ser. A, 64, 1993, 189-215, Theorem 9.4.
J. E. Iglesias, Enumeration of closest-packings by the space group: a simple approach, Z. Krist. 221 (2006) 237-245, eq. (5).

This array is very similar to but different from A011847.
Cf. A051168, A092964, A241926, A047996, A037306, A245559.
Rows include A001840, A006918, A051170, A011796, A011797, A031164. Main diagonal is A022553.

# To produce the first 10 rows and columns (as on page 174 of the Elashvili et al. 1999 reference):
with(numtheory):
cnk:=(n,k) -> add(mobius(n/d)*d, d in divisors(gcd(n,k)));
anmk:=(n,m,k)->(1/(n+m))*add( cnk(d,k)*binomial((n+m)/d,n/d), d in divisors(gcd(n,m))); # anmk(n,m,k) is the value of a_k(n,m) as in Theorem 1, Equation (4), of the Elashvili et al. 1999 reference.
r2:=(n,k)->[seq(anmk(n,m,k),m=1..10)];
for n from 1 to 10 do lprint(r2(n,1)); od:

rows = 12;
cnk[n_, k_] := Sum[MoebiusMu[n/d] d, {d , Divisors[GCD[n, k]]}];
anmk[n_, m_, k_] := (1/(n+m)) Sum[cnk[d, k] Binomial[(n+m)/d, n/d], {d, Divisors[GCD[n, m]]}];
r2[n_, k_] := Table[anmk[n, m, k], {m, 1, rows}];
T = Table[r2[n, 1], {n, 1, rows}];
Table[T[[n-k+1, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 05 2018, from Maple *)

A011795 a(n) = floor(C(n,4)/5).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 7, 14, 25, 42, 66, 99, 143, 200, 273, 364, 476, 612, 775, 969, 1197, 1463, 1771, 2125, 2530, 2990, 3510, 4095, 4750, 5481, 6293, 7192, 8184, 9275, 10472, 11781, 13209, 14763, 16450, 18278, 20254, 22386, 24682, 27150, 29799, 32637, 35673, 38916, 42375, 46060, 49980, 54145, 58565, 63250, 68211, 73458, 79002, 84854, 91025, 97527, 104371

Offset: 0

N. J. A. Sloane, David Broadhurst

a(n-1) = number of aperiodic necklaces (Lyndon words) with 5 black beads and n-5 white beads.

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See pp. 3, 11, 19.
Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Cf. A000031, A001037, A011915, A051168.
Same as A051170(n+1).
A column of triangle A011847.

[Floor(Binomial(n+1,5)/(n+1)): n in [0..70]]; // Vincenzo Librandi Jun 19 2012

seq(floor(binomial(n,4)/5), n=0.. 70); # Zerinvary Lajos, Jan 12 2009

CoefficientList[Series[x^5(1+x^3)/((1-x)^3(1-x^2)(1-x^5)),{x,0,70}],x] (* Vincenzo Librandi, Jun 19 2012 *)
CoefficientList[Series[x^4/5 (1/(1-x)^5-1/(1- x^5)),{x,0,70}],x] (* Herbert Kociemba, Oct 16 2016 *)

a(n)=binomial(n,4)\5 \\ Charles R Greathouse IV, Oct 07 2015

[binomial(n,4)//5 for n in range(71)] # G. C. Greubel, Oct 20 2024

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

a(n) = floor(binomial(n+1,5)/(n+1)). - Gary Detlefs, Nov 23 2011

A011797 a(n) = floor(C(n,6)/7).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 4, 12, 30, 66, 132, 245, 429, 715, 1144, 1768, 2652, 3876, 5537, 7752, 10659, 14421, 19228, 25300, 32890, 42287, 53820, 67860, 84825, 105183, 129456, 158224, 192129, 231880, 278256

Offset: 0

N. J. A. Sloane, David Broadhurst

a(n-1) is the number of aperiodic necklaces (Lyndon words) with 7 black beads and n-7 white beads.

David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1,1,-6,15,-20,15,-6,1).

Cf. A000031, A001037, A051168. Same as A051172(n+1).
First differences of A011853.
A column of triangle A011847.

CoefficientList[Series[x^6/7 (1/(1-x)^7-1/(1- x^7)),{x,0,40}],x]; (* Herbert Kociemba, Oct 16 2016 *)

a(n) = binomial(n, 6)\7; \\ Michel Marcus, Oct 16 2016

G.f.: (1+x^3)^2/((1-x)^4(1-x^2)^2(1-x^7))*x^7.
a(n) = floor(binomial(n+1,7)/(n+1)). [Gary Detlefs, Nov 23 2011]
G.f.: (x^6/7)*(1/(1-x)^7-1/(1- x^7)). - Herbert Kociemba, Oct 16 2016

A011842 a(n) = floor(n(n-1)(n-2)/24).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 8, 14, 21, 30, 41, 55, 71, 91, 113, 140, 170, 204, 242, 285, 332, 385, 442, 506, 575, 650, 731, 819, 913, 1015, 1123, 1240, 1364, 1496, 1636, 1785, 1942, 2109, 2284, 2470, 2665, 2870, 3085, 3311, 3547, 3795, 4053, 4324, 4606, 4900, 5206, 5525, 5856, 6201, 6558, 6930, 7315, 7714, 8127, 8555, 8997, 9455, 9927, 10416, 10920

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,1,-3,3,-1).

A column of triangle A011847.

Cf. A011886.

[Floor(Binomial(n,3)/4): n in [0..80]]; // G. C. Greubel, Oct 20 2024

seq(floor(binomial(n,3)/4), n=0..43); # Zerinvary Lajos, Jan 12 2009

Floor[Binomial[Range[0,80], 3]/4] (* G. C. Greubel, Oct 20 2024 *)

[binomial(n,3)//4 for n in range(81)] # G. C. Greubel, Oct 20 2024

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-8) - 3*a(n-9) + 3*a(n-10) - a(n-11).
G.f.: x^4*(1-x+x^2)*(1+x^2-x^3+x^4) / ((1-x)^4*(1+x)*(1+x^2)*(1+x^4)). (End)
a(n) = floor(binomial(n+1,4)/(n+1)). - Gary Detlefs, Nov 23 2011

More terms added by G. C. Greubel, Oct 20 2024

A032169 Number of aperiodic necklaces of n beads of 2 colors, 11 of them black.

Original entry on oeis.org

1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32065, 58786, 104006, 178296, 297160, 482885, 766935, 1193010, 1820910, 2731365, 4032015, 5864749, 8414640, 11920740, 16689036, 23107896, 31666376, 42975796

Offset: 12

Christian G. Bower

nonn

From Petros Hadjicostas, Aug 26 2018: (Start)
Assume n >= k >= 2. If a_k(n) is the number of aperiodic necklaces of n beads of 2 colors such that k of them are black and n-k of them are white, then a_k(n) = (1/k)*Sum_{d|gcd(n,k)} mu(d)*binomial(n/d - 1, k/d - 1) = (1/n)*Sum_{d|gcd(n,k)} mu(d)*binomial(n/d, k/d). This follows from Herbert Kociemba's general formula for the g.f. of (a_k(n): n>=1) that can be found in the comments for sequence A032168.
For k prime, we get a_k(n) = floor(binomial(n-1, k-1)/k). In such a case, the sequence becomes a column for triangle A011847. (This is not true when k is composite >= 4.)
(End)

Ray Chandler, Table of n, a(n) for n = 12..1012
C. G. Bower, Transforms (2)
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]
Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1,1,-10,45,-120,210,-252,210,-120,45,-10,1).

A column of triangle A011847.

CoefficientList[Series[x^11/11 (1/(1-x)^11-1/(1- x^11)),{x,0,50}],x] (* Herbert Kociemba, Oct 16 2016 *)

"CHK[ 11 ]" (necklace, identity, unlabeled, 11 parts) transform of 1, 1, 1, 1, ...
G.f.: (x^11/11)*(1/(1-x)^11-1/(1-x^11)). - Herbert Kociemba, Oct 16 2016
a(n) = (1/11)*(binomial(n-1, 10) - I(11|n)) = floor(binomial(n-1, 10)/11) for n >= 12, where I(a|b) = 1 if integer a divides integer b, and 0 otherwise. - Petros Hadjicostas, Aug 26 2018

A095718 a(n) = Sum_{k=0..n} floor(binomial(n,k)/(k+1)).

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 30, 56, 101, 186, 339, 630, 1167, 2182, 4092, 7710, 14561, 27594, 52425, 99862, 190647, 364722, 699045, 1342176, 2581107, 4971024, 9586975, 18512790, 35791386, 69273666, 134217720, 260301046, 505290269, 981706808

Offset: 1

Mike Zabrocki, Jul 08 2004

nonn

Row sums of A011847.

Robert Israel, Table of n, a(n) for n = 1..3329

Cf. A011847, A101687.

A095718:= func< n | (&+[Floor(Binomial(n,k)/(k+1)): k in [0..n]]) >;
[A095718(n): n in [1..40]]; // G. C. Greubel, Oct 20 2024

a:=n->add(floor(combinat[numbcomb](n,k)/(k+1)),k=0..n);

A095718[n_]:= Sum[Floor[Binomial[n,k]/(k+1)], {k,0,n}];
Table[A095718[n], {n,40}] (* G. C. Greubel, Oct 20 2024 *)

a(n) = sum(k=0, n, binomial(n,k)\(k+1)); \\ Michel Marcus, May 08 2018

def A095718(n): return sum(binomial(n,k)//(k+1) for k in range(n+1))
[A095718(n) for n in range(1,41)] # G. C. Greubel, Oct 20 2024

a(n) = Sum_{k=0..n} floor(binomial(n,k)/(k+1)).
From Robert Israel, May 07 2018: (Start)
(2^(n+1)-1)/(n+1) >= a(n) >= (2^(n+1)-1)/(n+1) - n.
It appears that a(n) = (2^(n+1)-2)/(n+1) if n+1 is prime. (End)

A011845 a(n) = floor( binomial(n,8)/9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 18, 55, 143, 333, 715, 1430, 2701, 4862, 8398, 13996, 22610, 35530, 54479, 81719, 120175, 173586, 246675, 345345, 476905, 650325, 876525, 1168700, 1542684, 2017356, 2615091, 3362260, 4289780, 5433721, 6835972, 8544965, 10616471

Offset: 0

N. J. A. Sloane

nonn

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. A000031, A001037, A028723, A051168.

A column of triangle A011847.

[Binomial(n, 8) div 9: n in [0..45]]; // Vincenzo Librandi, Oct 15 2015

Table[Floor[Binomial[n, 8] / 9], {n, 1, 50}] (* Vincenzo Librandi, Oct 15 2015 *)

a(n) = binomial(n, 8)\9; \\ Michel Marcus, Oct 14 2015

a(n) = floor(binomial(n+1,9)/(n+1)). [Gary Detlefs, Nov 23 2011]

Definition corrected by Pedro Antonio, Oct 14 2015

More terms from Vincenzo Librandi, Oct 15 2015

cp's OEIS Frontend

A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).

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A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.

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A011857 Triangle of numbers [ C(n,k)/k ], k=1..n-1.

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A245558 Square array read by antidiagonals: T(n,k) = number of n-tuples of nonnegative integers (u_0,...,u_{n-1}) satisfying Sum_{j=0..n-1} j*u_j == 1 mod n and Sum_{j=0..n-1} u_j = m.

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A011795 a(n) = floor(C(n,4)/5).

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A011797 a(n) = floor(C(n,6)/7).

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A011842 a(n) = floor(n*(n-1)*(n-2)/24).

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A011842 a(n) = floor(n(n-1)(n-2)/24).