A001401 -id:A001401 - cp's OEIS frontend

A008619 Positive integers repeated.

1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38

Offset: 0

N. J. A. Sloane

The floor of the arithmetic mean of the first n+1 positive integers. - Cino Hilliard, Sep 06 2003
Number of partitions of n into powers of 2 where no power is used more than three times, or 4th binary partition function (see A072170).
Number of partitions of n in which the greatest part is at most 2. - Robert G. Wilson v, Jan 11 2002
Number of partitions of n into at most 2 parts. - Jon Perry, Jun 16 2003
a(n) = #{k=0..n: k+n is even}. - Paul Barry, Sep 13 2003
Number of symmetric Dyck paths of semilength n+2 and having two peaks. E.g., a(6)=4 because we have UUUUUUU*DU*DDDDDDD, UUUUUU*DDUU*DDDDDD, UUUUU*DDDUUU*DDDDD and UUUU*DDDDUUUU*DDDD, where U=(1,1), D=(1,-1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004
Smallest positive integer whose harmonic mean with another positive integer is n (for n > 0). For example, a(6)=4 is already given (as 4 is the smallest positive integer such that the harmonic mean of 4 (with 12) is 6) - but the harmonic mean of 2 (with -6) is also 6 and 2 < 4, so the two positive integer restrictions need to be imposed to rule out both 2 and -6.
Second outermost diagonal of Losanitsch's triangle (A034851). - Alonso del Arte, Mar 12 2006
Arithmetic mean of n-th row of A080511. - Amarnath Murthy, Mar 20 2003
a(n) is the number of ways to pay n euros (or dollars) with coins of one and two euros (respectively dollars). - Richard Choulet and Robert G. Wilson v, Dec 31 2007
Inverse binomial transform of A045623. - Philippe Deléham, Dec 30 2008
Coefficient of q^n in the expansion of (m choose 2)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
This Itakura comment follows from a partial fraction decomposition (m choose 2)q = [(1-q^(2m-2))/(1+q) + (1-q^(2m-2))/(1-q) +2 (1-q^(m-1))^2/(1-q)^2]/4. Interpreted as generating functions in q, they have convolution structures; the first term in the numerator creates +1,-1,+1,-1 etc, the 2nd term creates +1,+1,+1,+1 etc., the 3rd term 2,4,6,8, etc. as m->infinity. - _R. J. Mathar, Sep 25 2008
Binomial transform of (-1)^n*A034008(n) = [1,0,1,-2,4,-8,16,-32,...]. - Philippe Deléham, Nov 15 2009
From Jon Perry_, Nov 16 2010: (Start)
Column sums of:
  1 1 1 1 1 1...
      1 1 1 1...
          1 1...
  ..............
  --------------
  1 1 2 2 3 3...  (End)
This sequence is also the half-convolution of the powers of 1 sequence A000012 with itself. For the definition of half-convolution see a comment on A201204, where also the rule for the o.g.f. is given. - Wolfdieter Lang, Jan 09 2012
a(n) is also the number of roots of the n-th Bernoulli polynomial in the right half-plane for n>0. - Michel Lagneau, Nov 08 2012
a(n) is the number of symmetry-allowed, linearly-independent terms at n-th order in the series expansion of the Exe vibronic perturbation matrix, H(Q) (cf. Viel & Eisfeld). - Bradley Klee, Jul 21 2015
a(n) is the number of distinct integers in the n-th row of Pascal's triangle. - Melvin Peralta, Feb 03 2016
a(n+1) for n >= 3 is the diameter of the Generalized Petersen Graph G(n, 1). - Nick Mayers, Jun 06 2016
The arithmetic function v_1(n,2) as defined in A289198. - Robert Price, Aug 22 2017
Also, this sequence is the second column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x. - Mohammad K. Azarian, Jul 18 2018
a(n+2) is the least k such that given any k integers, there exist two of them whose sum or difference is divisible by n. - Pablo Hueso Merino, May 09 2020
Column k = 2 of A051159. - John Keith, Jun 28 2021

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 109, Eq. [6c]; p. 116, P(n,2).
D. Parisse, 'The tower of Hanoi and the Stern-Brocot Array', Thesis, Munich 1997

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 120
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 209
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 351
Gerzson Keri and Patric R. J. Östergård, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.
L. F. Klosinski, G. L. Alexanderson and A. P. Hillman, The William Lowell Putnam Mathematical Competition, Amer. Math. Monthly 91 (1984), 487-495. See Problem B2.
Donatella Merlini and Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
Narad Rampersad and Max Wiebe, Sums of products of binomial coefficients mod 2 and 2-regular sequences, arXiv:2309.04012 [math.NT], 2023.
Bruce Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.
Alexandra Viel and Wolfgang Eisfeld, Effects of higher order Jahn-Teller coupling on the nuclear dynamics, J. Chem. Phys., 120, 4603 (2004).
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature
Index entries for sequences related to Stern's sequences
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Index entries for Molien series

Essentially same as A004526.
Harmonic mean of a(n) and A056136 is n.
Cf. A001057, A065033, A001399, A001400, A001401.
a(n)=A010766(n+2, 2).
Cf. A010551 (partial products).
Cf. A263997 (a block spiral).
Cf. A289187.
Column 2 of A235791.

a008619 = (+ 1) . (`div` 2)
a008619_list = concatMap (\x -> [x,x]) [1..]
-- Reinhard Zumkeller, Apr 02 2012

I:=[1,1,2]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..100]]; // Vincenzo Librandi, Feb 04 2015

a:= n-> iquo(n+2, 2): seq(a(n), n=0..75);

Flatten[Table[{n,n},{n,35}]] (* Harvey P. Dale, Sep 20 2011 *)
With[{c=Range[40]},Riffle[c,c]] (* Harvey P. Dale, Feb 23 2013 *)
CoefficientList[Series[1/(1 - x - x^2 + x^3), {x, 0, 75}], x] (* Robert G. Wilson v, Feb 05 2015 *)
LinearRecurrence[{1, 1, -1}, {1, 1, 2}, 75] (* Robert G. Wilson v, Feb 05 2015 *)
Table[QBinomial[n, 2, -1], {n, 2, 75}] (* John Keith, Jun 28 2021 *)

PARI
```
a(n)=n\2+1
```

def A008619(n): return (n>>1)+1 # Chai Wah Wu, Jul 07 2022

a = lambda n: 1 if n==0 else a(n-1)+1 if 2.divides(n) else a(n-1) # Peter Luschny, Feb 05 2015

(2 to 99).map( / 2) // _Alonso del Arte, May 09 2020

Euler transform of [1, 1].
a(n) = 1 + floor(n/2).
G.f.: 1/((1-x)(1-x^2)).
E.g.f.: ((3+2*x)*exp(x) + exp(-x))/4.
a(n) = a(n-1) + a(n-2) - a(n-3) = -a(-3-n).
a(0) = a(1) = 1 and a(n) = floor( (a(n-1) + a(n-2))/2 + 1 ).
a(n) = (2*n + 3 + (-1)^n)/4. - Paul Barry, May 27 2003
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} binomial(j, i)*(-2)^i. - Paul Barry, Aug 26 2003
E.g.f.: ((1+x)*exp(x) + cosh(x))/2. - Paul Barry, Sep 13 2003
a(n) = A108299(n-1,n)*(-1)^floor(n/2) for n > 0. - Reinhard Zumkeller, Jun 01 2005
a(n) = A108561(n+2,n) for n > 0. - Reinhard Zumkeller, Jun 10 2005
a(n) = A125291(A125293(n)) for n>0. - Reinhard Zumkeller, Nov 26 2006
a(n) = ceiling(n/2), n >= 1. - Mohammad K. Azarian, May 22 2007
INVERT transformation yields A006054 without leading zeros. INVERTi transformation yields negative of A124745 with the first 5 terms there dropped. - R. J. Mathar, Sep 11 2008
a(n) = A026820(n,2) for n > 1. - Reinhard Zumkeller, Jan 21 2010
a(n) = n - a(n-1) + 1 (with a(0)=1). - Vincenzo Librandi, Nov 19 2010
a(n) = A000217(n) / A110654(n). - Reinhard Zumkeller, Aug 24 2011
a(n+1) = A181971(n,n). - Reinhard Zumkeller, Jul 09 2012
1/(1+2/(2+3/(3+4/(4+5/(5+...(continued fraction))))) = 1/(e-1), see A073333. - Philippe Deléham, Mar 09 2013
a(n) = floor(A000217(n)/n), n > 0. - L. Edson Jeffery, Jul 26 2013
a(n) = n*a(n-1) mod (n+1) = -a(n-1) mod (n+1), the least positive residue modulo n+1 for each expression for n > 0, with a(0) = 1 (basically restatements of Vincenzo Librandi's formula). - Rick L. Shepherd, Apr 02 2014
a(n) = (a(0) + a(1) + ... + a(n-1))/a(n-1), where a(0) = 1. - Melvin Peralta, Jun 16 2015
a(n) = Sum_{k=0..n} (-1)^(n-k) * (k+1). - Rick L. Shepherd, Sep 18 2020
a(n) = a(n-2) + 1 for n >= 2. - Vladimír Modrák, Sep 29 2020
a(n) = A004526(n)+1. - Chai Wah Wu, Jul 07 2022

Additional remarks from Daniele Parisse
Edited by N. J. A. Sloane, Sep 06 2009
Partially edited by Joerg Arndt, Mar 11 2010

A001399 a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.

Original entry on oeis.org

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, 300, 310, 320, 331, 341

Offset: 0

N. J. A. Sloane

Also number of tripods (trees with exactly 3 leaves) on n vertices. - Eric W. Weisstein, Mar 05 2011
Also number of partitions of n+3 into exactly 3 parts; number of partitions of n in which the greatest part is less than or equal to 3; and the number of nonnegative solutions to b + 2c + 3d = n.
Also a(n) gives number of partitions of n+6 into 3 distinct parts and number of partitions of 2n+9 into 3 distinct and odd parts, e.g., 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3. - Jon Perry, Jan 07 2004
Also bracelets with n+3 beads 3 of which are red (so there are 2 possibilities with 5 beads).
More generally, the number of partitions of n into at most k parts is also the number of partitions of n+k into k positive parts, the number of partitions of n+k in which the greatest part is k, the number of partitions of n in which the greatest part is less than or equal to k, the number of partitions of n+k(k+1)/2 into exactly k distinct positive parts, the number of nonnegative solutions to b + 2c + 3d + ... + kz = n and the number of nonnegative solutions to 2c + 3d + ... + kz <= n. - Henry Bottomley, Apr 17 2001
Also coefficient of q^n in the expansion of (m choose 3)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
From Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) for n > 0 is formed by the folding points (including the initial 1). The spiral begins:
.
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
       90  60  36  18   6   0   3  12  27  48  75
       /   /   /   /   /   /   /   /   /   /   /
     91  61  37  19   7   1---2  11  26  47  74
           \   \   \   \         /   /   /   /
           62  38  20   8---9--10  25  46  73
             \   \   \             /   /   /
             63  39  21--22--23--24  45  72
               \   \                 /   /
               64  40--41--42--43--44  71
                 \                     /
                 65--66--67--68--69--70
.
a(p) is maximal number of hexagons in a polyhex with perimeter at most 2p + 6. (End)
a(n-3) is the number of partitions of n into 3 distinct parts, where 0 is allowed as a part. E.g., at n=9, we can write 8+1+0, 7+2+0, 6+3+0, 4+5+0, 1+2+6, 1+3+5 and 2+3+4, which is a(6)=7. - Jon Perry, Jul 08 2003
a(n) gives number of partitions of n+6 into parts <=3 where each part is used at least once (subtract 6=1+2+3 from n). - Jon Perry, Jul 03 2004
This is also the number of partitions of n+3 into exactly 3 parts (there is a 1-to-1 correspondence between the number of partitions of n+3 in which the greatest part is 3 and the number of partitions of n+3 into exactly three parts). - Graeme McRae, Feb 07 2005
Apply the Riordan array (1/(1-x^3),x) to floor((n+2)/2). - Paul Barry, Apr 16 2005
Also, number of triangles that can be created with odd perimeter 3,5,7,9,11,... with all sides whole numbers. Note that triangles with even perimeter can be generated from the odd ones by increasing each side by 1. E.g., a(1) = 1 because perimeter 3 can make {1,1,1} 1 triangle. a(4) = 3 because perimeter 9 can make {1,4,4} {2,3,4} {3,3,3} 3 possible triangles. - Bruce Love (bruce_love(AT)ofs.edu.sg), Nov 20 2006
Also number of nonnegative solutions of the Diophantine equation x+2*y+3*z=n, cf. Pólya/Szegő reference.
From Vladimir Shevelev, Apr 23 2011: (Start)
Also a(n-3), n >= 3, is the number of non-equivalent necklaces of 3 beads each of them painted by one of n colors.
The sequence {a(n-3), n >= 3} solves the so-called Reis problem about convex k-gons in case k=3 (see our comment to A032279).
a(n-3) (n >= 3) is an essentially unimprovable upper estimate for the number of distinct values of the permanent in (0,1)-circulants of order n with three 1's in every row. (End)
A001399(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w = 2*x+3*y. - Clark Kimberling, Jun 04 2012
Also, for n >= 3, a(n-3) is the number of the distinct triangles in an n-gon, see the Ngaokrajang links. - Kival Ngaokrajang, Mar 16 2013
Also, a(n) is the total number of 5-curve coin patterns (5C4S type: 5 curves covering full 4 coins and symmetry) packing into fountain of coins base (n+3). See illustration in links. - Kival Ngaokrajang, Oct 16 2013
Also a(n) = half the number of minimal zero sequences for Z_n of length 3 [Ponomarenko]. - N. J. A. Sloane, Feb 25 2014
Also, a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of an Octahedral Rotational Energy Surface (cf. Harter & Patterson). - Bradley Klee, Jul 31 2015
Also Molien series for invariants of finite Coxeter groups D_3 and A_3. - N. J. A. Sloane, Jan 10 2016
Number of different distributions of n+6 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Jan 11 2016
a(n) is also the number of partitions of 2*n with <= n parts and no part >= 4. The bijection to partitions of n with no part >= 4 is: 1 <-> 2, 2 <-> 1 + 3, 3 <-> 3 + 3 (observing the order of these rules). The <- direction uses the following fact for partitions of 2*n with <= n parts and no part >=4: for each part 1 there is a part 3, and an even number (including 0) of remaining parts 3. - Wolfdieter Lang, May 21 2019
List of the terms in A000567(n>=1), A049450(n>=1), A033428(n>=1), A049451(n>=1), A045944(n>=1), and A003215(n) in nondecreasing order. List of the numbers A056105(n)-1, A056106(n)-1, A056107(n)-1, A056108(n)-1, A056109(n)-1, and A003215(m) with n >= 1 and m >= 0 in nondecreasing order. Numbers of the forms 3n*(n-1)+1, n*(3n-2), n*(3n-1), 3n^2, n*(3n+1), n*(3n+2) with n >= 1 listed in nondecreasing order. Integers m such that lattice points from 1 through m on a hexagonal spiral starting at 1 forms a convex polygon. - Ya-Ping Lu, Jan 24 2024

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 10*x^8 + 12*x^9 + ...
Recall that in a necklace the adjacent beads have distinct colors. Suppose we have n colors with labels 1,...,n. Two colorings of the beads are equivalent if the cyclic sequences of the distances modulo n between labels of adjacent colors have the same period. If n=4, all colorings are equivalent. E.g., for the colorings {1,2,3} and {1,2,4} we have the same period {1,1,2} of distances modulo 4. So, a(n-3)=a(1)=1. If n=5, then we have two such periods {1,1,3} and {1,2,2} modulo 5. Thus a(2)=2. - _Vladimir Shevelev_, Apr 23 2011
a(0) = 1, i.e., {1,2,3} Number of different distributions of 6 identical balls to 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Jan 11 2016
a(3) = 3, i.e., {1,2,6}, {1,3,5}, {2,3,4} Number of different distributions of 9 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Jan 11 2016
From _Gus Wiseman_, Apr 15 2019: (Start)
The a(0) = 1 through a(8) = 10 integer partitions of n with at most three parts are the following. The Heinz numbers of these partitions are given by A037144.
  ()  (1)  (2)   (3)    (4)    (5)    (6)    (7)    (8)
           (11)  (21)   (22)   (32)   (33)   (43)   (44)
                 (111)  (31)   (41)   (42)   (52)   (53)
                        (211)  (221)  (51)   (61)   (62)
                               (311)  (222)  (322)  (71)
                                      (321)  (331)  (332)
                                      (411)  (421)  (422)
                                             (511)  (431)
                                                    (521)
                                                    (611)
The a(0) = 1 through a(7) = 8 integer partitions of n + 3 whose greatest part is 3 are the following. The Heinz numbers of these partitions are given by A080193.
  (3)  (31)  (32)   (33)    (322)    (332)     (333)      (3322)
             (311)  (321)   (331)    (3221)    (3222)     (3331)
                    (3111)  (3211)   (3311)    (3321)     (32221)
                            (31111)  (32111)   (32211)    (33211)
                                     (311111)  (33111)    (322111)
                                               (321111)   (331111)
                                               (3111111)  (3211111)
                                                          (31111111)
Non-isomorphic representatives of the a(0) = 1 through a(5) = 5 unlabeled multigraphs with 3 vertices and n edges are the following.
  {}  {12}  {12,12}  {12,12,12}  {12,12,12,12}  {12,12,12,12,12}
            {13,23}  {12,13,23}  {12,13,23,23}  {12,13,13,23,23}
                     {13,23,23}  {13,13,23,23}  {12,13,23,23,23}
                                 {13,23,23,23}  {13,13,23,23,23}
                                                {13,23,23,23,23}
The a(0) = 1 through a(8) = 10 strict integer partitions of n - 6 with three parts are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A007304.
  (321)  (421)  (431)  (432)  (532)  (542)  (543)  (643)   (653)
                (521)  (531)  (541)  (632)  (642)  (652)   (743)
                       (621)  (631)  (641)  (651)  (742)   (752)
                              (721)  (731)  (732)  (751)   (761)
                                     (821)  (741)  (832)   (842)
                                            (831)  (841)   (851)
                                            (921)  (931)   (932)
                                                   (A21)   (941)
                                                           (A31)
                                                           (B21)
The a(0) = 1 through a(8) = 10 integer partitions of n + 3 with three parts are the following. The Heinz numbers of these partitions are given by A014612.
  (111)  (211)  (221)  (222)  (322)  (332)  (333)  (433)  (443)
                (311)  (321)  (331)  (422)  (432)  (442)  (533)
                       (411)  (421)  (431)  (441)  (532)  (542)
                              (511)  (521)  (522)  (541)  (551)
                                     (611)  (531)  (622)  (632)
                                            (621)  (631)  (641)
                                            (711)  (721)  (722)
                                                   (811)  (731)
                                                          (821)
                                                          (911)
The a(0) = 1 through a(8) = 10 integer partitions of n whose greatest part is <= 3 are the following. The Heinz numbers of these partitions are given by A051037.
  ()  (1)  (2)   (3)    (22)    (32)     (33)      (322)      (332)
           (11)  (21)   (31)    (221)    (222)     (331)      (2222)
                 (111)  (211)   (311)    (321)     (2221)     (3221)
                        (1111)  (2111)   (2211)    (3211)     (3311)
                                (11111)  (3111)    (22111)    (22211)
                                         (21111)   (31111)    (32111)
                                         (111111)  (211111)   (221111)
                                                   (1111111)  (311111)
                                                              (2111111)
                                                              (11111111)
The a(0) = 1 through a(6) = 7 strict integer partitions of 2n+9 with 3 parts, all of which are odd, are the following. The Heinz numbers of these partitions are given by A307534.
  (5,3,1)  (7,3,1)  (7,5,1)  (7,5,3)   (9,5,3)   (9,7,3)   (9,7,5)
                    (9,3,1)  (9,5,1)   (9,7,1)   (11,5,3)  (11,7,3)
                             (11,3,1)  (11,5,1)  (11,7,1)  (11,9,1)
                                       (13,3,1)  (13,5,1)  (13,5,3)
                                                 (15,3,1)  (13,7,1)
                                                           (15,5,1)
                                                           (17,3,1)
The a(0) = 1 through a(8) = 10 strict integer partitions of n + 3 with 3 parts where 0 is allowed as a part (A = 10):
  (210)  (310)  (320)  (420)  (430)  (530)  (540)  (640)  (650)
                (410)  (510)  (520)  (620)  (630)  (730)  (740)
                       (321)  (610)  (710)  (720)  (820)  (830)
                              (421)  (431)  (810)  (910)  (920)
                                     (521)  (432)  (532)  (A10)
                                            (531)  (541)  (542)
                                            (621)  (631)  (632)
                                                   (721)  (641)
                                                          (731)
                                                          (821)
The a(0) = 1 through a(7) = 7 integer partitions of n + 6 whose distinct parts are 1, 2, and 3 are the following. The Heinz numbers of these partitions are given by A143207.
  (321)  (3211)  (3221)   (3321)    (32221)    (33221)     (33321)
                 (32111)  (32211)   (33211)    (322211)    (322221)
                          (321111)  (322111)   (332111)    (332211)
                                    (3211111)  (3221111)   (3222111)
                                               (32111111)  (3321111)
                                                           (32211111)
                                                           (321111111)
(End)
Partitions of 2*n with <= n parts and no part >= 4: a(3) = 3 from (2^3), (1,2,3), (3^2) mapping to (1^3), (1,2), (3), the partitions of 3 with no part >= 4, respectively. - _Wolfdieter Lang_, May 21 2019

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III, Problem 33.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 110, D(n); page 263, #18, P_n^{3}.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
J. H. van Lint, Combinatorial Seminar Eindhoven, Lecture Notes Math., 382 (1974), see pp. 33-34.
G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, Sect. 1, Problem 25.
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).

Marius A. Burtea, Table of n, a(n) for n = 0..17501 (terms 0..1000 from T. D. Noe, terms 14001 onwards corrected by Sean A. Irvine, April 25 2019)
Hamid Afshar, Branislav Cvetkovic, Sabine Ertl, Daniel Grumiller, and Niklas Johansson, Conformal Chern-Simons holography-lock, stock and barrel, arXiv preprint arXiv:1110.5644 [hep-th], 2011.
C. Ahmed, P. Martin, and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68 (2013), 71-79.
N. Benyahia Tani, Z. Yahi, and S. Bouroubi, Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular n-gon, Bulletin du Laboratoire Liforce 01 (2014), 1-9.
Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal Systems Including the Corannulene and Coronene Homologs: Novel Applications of Pólya's Theorem, Z. Naturforsch., 52a (1997), 867-873.
Lucia De Luca and Gero Friesecke, Classification of particle numbers with unique Heitmann-Radin minimizer, arXiv:1701.07231 [math-ph], 2017.
Nick Fischer and Christian Ikenmeyer, The Computational Complexity of Plethysm Coefficients, arXiv:2002.00788 [cs.CC], 2020.
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
W. G. Harter and C. W. Patterson, Asymptotic eigensolutions of fourth and sixth rank octahedral tensor operators, Journal of Mathematical Physics, 20.7 (1979), 1453-1459. alternate copy
M. D. Hirschhorn and J. A. Sellers, Enumeration of unigraphical partitions, JIS 11 (2008) 08.4.6.
R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 352.
J. H. Jordan, R. Walch, and R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly 86 (1979), 686-689.
Alexander V. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.
Gerzson Keri and Patric R. J. Östergård, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences 9 (2006), Article 06.4.7.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Axel Kleinschmidt and Valentin Verschinin, Tetrahedral modular graph functions, J. High Energy Phys. 2017, No. 9, Paper No. 155, 38 p. (2017), eq (3.40).
Mathematics Stack Exchange, What does "pcr" stand for. [This is Comtet's notation for "prime circulator". See pp. 109-110.]
M. B. Nathanson, Partitions with parts in a finite set, arXiv:math/0002098 [math.NT], 2000.
Kival Ngaokrajang, Distinct triangles in n-gon for n = 3..9, Distinct triangles in 45-gon
Kival Ngaokrajang, Illustration of 5-curves coins patterns
Andrew N. Norris, Higher derivatives and the inverse derivative of a tensor-valued function of a tensor, arXiv:0707.0115 [math.SP], 2007; Equation 3.28, p. 10.
Jon Perry, More Partition Function.
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.
Vadim Ponomarenko, Minimal zero sequences of finite cyclic groups, INTEGERS 4 (2004), #A24.
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math. 35(5) (2004), 629-638.
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).
Devansh Singh and S. N. Mishra, Representing K-parts Integer Partitions, Global Journal of Pure and Applied Mathematics (GJPAM), Volume 15 Number 6 (2019), pp. 889-907.
Karl Hermann Struve, Fresnel's Interferenzerscheinungen: Theoretisch und Experimentell Bearbeitet, Dorpat, 1881 (Thesis). [Gives the Round(n^2/12) formula.]
James Tanton, Young students approach integer triangles, FOCUS 22(5) (2002), 4 - 6.
James Tanton, Integer Triangles, Chapter 11 in "Mathematics Galore!" (MAA, 2012).
James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See p. 11.
Richard Vale and Shayne Waldron, The construction of G-invariant finite tight frames, J. Four. Anal. Applic. 22 (2016), 1097-1120.
Eric Weisstein's World of Mathematics, Tripod.
Gus Wiseman, Number of ordered triples of distinct positive integers summing to n.
Gus Wiseman, Number of non-unimodal triples of distinct positive integers summing to n.
Gus Wiseman, Number of unimodal triples of distinct positive integers summing to n.
Winston C. Yang, Maximal and minimal polyhexes, 2002.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).
Index entries for Molien series.
Index entries for two-way infinite sequences

Cf. A008724, A003082, A117485, A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228, A036496, A008619, A001400, A001401, A069905, A008615, row 3 of A192517.
Molien series for finite Coxeter groups D_3 through D_12 are A001399, A051263, A266744-A266751.
Cf. A007304, A014612, A037144, A051037, A080193, A143207, A307534.

a001399 = p [1,2,3] where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 28 2013

I:=[1,1,2,3,4,5]; [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..80]]; // Vincenzo Librandi, Feb 14 2015

[#RestrictedPartitions(n,{1,2,3}): n in [0..62]]; // Marius A. Burtea, Jan 06 2019

[Round((n+3)^2/12): n in [0..70]]; // Marius A. Burtea, Jan 06 2019

A001399 := proc(n)
    round( (n+3)^2/12) ;
end proc:
seq(A001399(n),n=0..40) ;
with(combstruct):ZL4:=[S,{S=Set(Cycle(Z,card<4))}, unlabeled]:seq(count(ZL4,size=n),n=0..61); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=3)},unlabelled]: seq(combstruct[count](B, size=n), n=0..61); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)), {x, 0, 65} ], x ]
Table[ Length[ IntegerPartitions[n, 3]], {n, 0, 61} ] (* corrected by Jean-François Alcover, Aug 08 2012 *)
k = 3; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
LinearRecurrence[{1,1,0,-1,-1,1},{1,1,2,3,4,5},70] (* Harvey P. Dale, Jun 21 2012 *)
a[ n_] := With[{m = Abs[n + 3] - 3}, Length[ IntegerPartitions[ m, 3]]]; (* Michael Somos, Dec 25 2014 *)
k=3 (* Number of red beads in bracelet problem *);CoefficientList[Series[(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2,{x,0,50}],x] (* Herbert Kociemba, Nov 04 2016 *)
Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Apr 15 2019 *)

{a(n) = round((n + 3)^2 / 12)}; /* Michael Somos, Sep 04 2006 */

[round((n+3)**2 / 12) for n in range(0,62)] # Ya-Ping Lu, Jan 24 2024

G.f.: 1/((1 - x) * (1 - x^2) * (1 - x^3)) = -1/((x+1)*(x^2+x+1)*(x-1)^3); Simon Plouffe in his 1992 dissertation
a(n) = round((n + 3)^2/12). Note that this cannot be of the form (2*i + 1)/2, so ties never arise.
a(n) = A008284(n+3, 3), n >= 0.
a(n) = 1 + a(n-2) + a(n-3) - a(n-5) for all n in Z. - Michael Somos, Sep 04 2006
a(n) = a(-6 - n) for all n in Z. - Michael Somos, Sep 04 2006
a(6*n) = A003215(n), a(6*n + 1) = A000567(n + 1), a(6*n + 2) = A049450(n + 1), a(6*n + 3) = A033428(n + 1), a(6*n + 4) = A049451(n + 1), a(6*n + 5) = A045944(n + 1).
a(n) = a(n-1) + A008615(n+2) = a(n-2) + A008620(n) = a(n-3) + A008619(n) = A001840(n+1) - a(n-1) = A002620(n+2) - A001840(n) = A000601(n) - A000601(n-1). - Henry Bottomley, Apr 17 2001
P(n, 3) = (1/72) * (6*n^2 - 7 - 9*pcr{1, -1}(2, n) + 8*pcr{2, -1, -1}(3, n)) (see Comtet). [Here "pcr" stands for "prime circulator" and it is defined on p. 109 of Comtet, while the formula appears on p. 110. - Petros Hadjicostas, Oct 03 2019]
Let m > 0 and -3 <= p <= 2 be defined by n = 6*m+p-3; then for n > -3, a(n) = 3*m^2 + p*m, and for n = -3, a(n) = 3*m^2 + p*m + 1. - Floor van Lamoen, Jul 23 2001
72*a(n) = 17 + 6*(n+1)*(n+5) + 9*(-1)^n - 8*A061347(n). - Benoit Cloitre, Feb 09 2003
From Jon Perry, Jun 17 2003: (Start)
a(n) = 6*t(floor(n/6)) + (n%6) * (floor(n/6) + 1) + (n mod 6 == 0?1:0), where t(n) = n*(n+1)/2.
a(n) = ceiling(1/12*n^2 + 1/2*n) + (n mod 6 == 0?1:0).
[Here "n%6" means "n mod 6" while "(n mod 6 == 0?1:0)" means "if n mod 6 == 0 then 1, else 0" (as in C).]
(End)
a(n) = Sum_{i=0..floor(n/3)} 1 + floor((n - 3*i)/2). - Jon Perry, Jun 27 2003
a(n) = Sum_{k=0..n} floor((k + 2)/2) * (cos(2*Pi*(n - k)/3 + Pi/3)/3 + sqrt(3) * sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3). - Paul Barry, Apr 16 2005
(m choose 3)_q = (q^m-1) * (q^(m-1) - 1) * (q^(m-2) - 1)/((q^3 - 1) * (q^2 - 1) * (q - 1)).
a(n) = Sum_{k=0..floor(n/2)} floor((3 + n - 2*k)/3). - Paul Barry, Nov 11 2003
A117220(n) = a(A003586(n)). - Reinhard Zumkeller, Mar 04 2006
a(n) = 3 * Sum_{i=2..n+1} floor(i/2) - floor(i/3). - Thomas Wieder, Feb 11 2007
Identical to the number of points inside or on the boundary of the integer grid of {I, J}, bounded by the three straight lines I = 0, I - J = 0 and I + 2J = n. - Jonathan Vos Post, Jul 03 2007
a(n) = A026820(n,3) for n > 2. - Reinhard Zumkeller, Jan 21 2010
Euler transform of length 3 sequence [ 1, 1, 1]. - Michael Somos, Feb 25 2012
a(n) = A005044(2*n + 3) = A005044(2*n + 6). - Michael Somos, Feb 25 2012
a(n) = A000212(n+3) - A002620(n+3). - Richard R. Forberg, Dec 08 2013
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6). - David Neil McGrath, Feb 14 2015
a(n) = floor((n^2+3)/12) + floor((n+2)/2). - Giacomo Guglieri, Apr 02 2019
From Devansh Singh, May 28 2020: (Start)
Let p(n, 3) be the number of 3-part integer partitions in which every part is > 0.
Then for n >= 3, p(n, 3) is equal to:
  (n^2 - 1)/12 when n is odd and 3 does not divide n.
  (n^2 + 3)/12 when n is odd and 3 divides n.
  (n^2 - 4)/12 when n is even and 3 does not divide n.
  (n^2)/12 when n is even and 3 divides n.
For n >= 3, p(n, 3) = a(n-3). (End)
a(n) = floor(((n+3)^2 + 4)/12). - Vladimír Modrák, Zuzana Soltysova, Dec 08 2020
Sum_{n>=0} 1/a(n) = 15/4 - Pi/(2*sqrt(3)) + Pi^2/18 + tanh(Pi/(2*sqrt(3)))*Pi/sqrt(3). - Amiram Eldar, Sep 29 2022
E.g.f.: exp(-x)*(9 + exp(2*x)*(47 + 42*x + 6*x^2) + 16*exp(x/2)*cos(sqrt(3)*x/2))/72. - Stefano Spezia, Mar 05 2023
a(6n) = 1+6*A000217(n); Sum_{i=1..n} a(6*i) = A000578(n+1). - David García Herrero, May 05 2024

Name edited by Gus Wiseman, Apr 15 2019

A001400 Number of partitions of n into at most 4 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632, 672, 720, 764, 816, 864, 920, 972, 1033, 1089, 1154, 1215, 1285, 1350, 1425, 1495

Offset: 0

N. J. A. Sloane

Molien series for 4-dimensional representation of S_4 [Nebe, Rains, Sloane, Chap. 7].
Also number of pure 2-complexes on 4 nodes with n multiple 2-simplexes. - Vladeta Jovovic, Dec 27 1999
Also number of different integer triangles with perimeter <= n+3. Also number of different scalene integer triangles with perimeter <= n+9. - Reinhard Zumkeller, May 12 2002
a(n) is the coefficient of q^n in the expansion of (m choose 4)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
Also number of partitions of n into parts <= 4. a(n) = A026820(n,4), for n > 3. - Reinhard Zumkeller, Jan 21 2010
Number of different distributions of n+10 identical balls in 4 boxes as x,y,z,p where 0 < x < y < z < p. - Ece Uslu and Esin Becenen, Jan 11 2016
Number of partitions of 5n+8 or 5n+12 into 4 parts (+-) 3 mod 5. a(4) = 5 partitions of 28: [7,7,7,7], [12,7,7,2], [12,12,2,2], [17,7,2,2], [22,2,2,2]. a(3) = 3 partitions of 27: [8,8,8,3], [13,8,3,3], [18,3,3,3]. - Richard Turk, Feb 24 2016
a(n) is the total number of non-isomorphic geodetic graphs of diameter n homeomorphic to a complete graph K4. - Carlos Enrique Frasser, May 24 2018

			(4 choose 4)_q = 1, (5 choose 4)_q = q^4 + q^3 + q^2 + q + 1, (6 choose 4)_q = q^8 + q^7 + 2*q^6 + 2*q^5 + 3*q^4 + 2*q^3 + 2*q^2 + q + 1, (7 choose 4) = q^12 + q^11 + 2*q^10 + 3*q^9 + 4*q^8 + 4*q^7 + 5*q^6 + 4*q^5 + 4*q^4 + 3*q^3 + 2*q^2 + q + 1 so the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2 and so on.
G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 9*x^6 + 11*x^7 + ...
a(4) = 5, i.e., {1,2,3,8}, {1,2,4,7}, {1,2,5,6}, {2,3,4,5}, {1,3,4,6}. Number of different distributions of 14 identical balls in 4 boxes as x,y,z,p where 0 < x < y < z < p. - _Ece Uslu_, Esin Becenen, Jan 11 2016

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=4 of Q(m,n) table; p. 120, P(n,4).
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
D. E. Knuth, The Art of Computer Programming, vol. 4, Fascicle 3, Generating All Combinations and Partitions, Addison-Wesley, 2005, Section 7.2.1.4., p. 56, exercise 31.
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
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon. Rostocker Math. Kolloq. 68, 71-79 (2013), g(Z).
Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
V. M. Buchstaber and A. V. Ustinov, Coefficient rings of formal group laws, Sbornik: Mathematics, Volume 206, Number 11.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Éva Czabarka, Peter Dankelmann, Trevor Olsen, and László A. Székely, Wiener Index and Remoteness in Triangulations and Quadrangulations, arXiv:1905.06753 [math.CO], 2019.
F. Ellermann, Illustration for A001400, A061924
C. E. Frasser and G. N. Vostrov, Geodetic Graphs Homeomorphic to a Given Geodetic Graph, arXiv:1611.01873 [cs.DM], 2016. [p. 16, corollary 5]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 353
Gerzson Keri and Patric R. J. Östergård, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Jon Perry, More Partition Functions
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
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).
Index entries for two-way infinite sequences

Essentially same as A026810. Partial sums of A005044.
a(n) = A008284(n+4, 4), n >= 0.
Cf. A008619, A001399, A001401, A026820, A070083, A117486.
First differences of A002621.

a001400 n = a001400_list !! n
a001400_list = scanl1 (+) a005044_list -- Reinhard Zumkeller, Feb 28 2013

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; G:=MatrixGroup<4,K|q1,q2,h>; MolienSeries(G);

A001400 := n->if n mod 2 = 0 then round(n^2*(n+3)/144); else round((n-1)^2*(n+5)/144); fi;
with(combstruct):ZL5:=[S,{S=Set(Cycle(Z,card<5))}, unlabeled]:seq(count(ZL5,size=n),n=0..55); # Zerinvary Lajos, Sep 24 2007
A001400:=-(-z**8+z**9+2*z**4-z**7-1-z)/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**4; # [conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for an initial 1]
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=4)},unlabelled]: seq(combstruct[count](B, size=n), n=0..55); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 65} ], x ]
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {1, 1, 2, 3, 5, 6, 9, 11, 15, 18}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
a[n_] := Sum[Floor[(n - j - 3*k + 2)/2], {j, 0, Floor[n/4]}, {k, j, Floor[(n - j)/3]}]; Table[a[n], {n, 0, 55}] (* L. Edson Jeffery, Jul 31 2014 *)
a[ n_] := With[{m = n + 5}, Round[ (2 m^3 - 3 m (5 + 3 (-1)^m)) / 288]]; (* Michael Somos, Dec 29 2014 *)
a[ n_] := With[{m = Abs[n + 5] - 5}, Sign[n + 5] Length[ IntegerPartitions[ m, 4]]]; (* Michael Somos, Dec 29 2014 *)
a[ n_] := With[{m = Abs[n + 5] - 5}, Sign[n + 5] SeriesCoefficient[ 1 / ((1 - x) (1 - x^2) (1 - x^3) (1 - x^4)), {x, 0, m}]]; (* Michael Somos, Dec 29 2014 *)
Table[Length@IntegerPartitions[n, 4], {n, 0, 55}] (* Robert Price, Aug 18 2020 *)

a(n) = round(((n+4)^3 + 3*(n+4)^2 -9*(n+4)*((n+4)% 2))/144) \\ Washington Bomfim, Jul 03 2012

{a(n) = n+=5; round( (2*n^3 - 3*n*(5 + 3*(-1)^n)) / 288)}; \\ Michael Somos, Dec 29 2014

a(n) = #partitions(n,,4); \\ Ruud H.G. van Tol, Jun 02 2024

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
a(n) = 1 + (a(n-2) + a(n-3) + a(n-4)) - (a(n-5) + a(n-6) + a(n-7)) + a(n-9). - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
P(n, 4) = (1/288)*(2*n^3 + 6*n^2 - 9*n - 13 + (9*n+9)*pcr{1, -1}(2, n) - 32*pcr{1, -1, 0}(3, n) - 36*pcr{1, 0, -1, 0}(4, n)) (see Comtet).
Let c(n) = Sum_{i=0..floor(n/3)} (1 + ceiling((n-3*i-1)/2)), then a(n) = Sum_{i=0..floor(n/4)} (1 + ceiling((n-4*i-1)/2) + c(n-4*i-3)). - Jon Perry, Jun 27 2003
Euler transform of finite sequence [1, 1, 1, 1].
(n choose 4)_q = (q^n-1)*(q^(n-1)-1)*(q^(n-2)-1)*(q^(n-3)-1)/((q^4-1)*(q^3-1)*(q^2-1)*(q-1)).
a(n) = round(((n+4)^3 + 3*(n+4)^2 - 9*(n+4)*((n+4) mod 2))/144). - Washington Bomfim, Jul 03 2012
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10). - David Neil McGrath, Sep 12 2014
a(n) = -a(-10-n) for all n in Z. - Michael Somos, Dec 29 2014
a(n) - a(n+1) - a(n+3) + a(n+4) = 0 if n is odd, else floor(n/4) + 2 for all n in Z. - Michael Somos, Dec 29 2014
a(n) = n^3/144 + n^2/24 - 7*n/144 + 1 + floor(n/4)/4 + floor(n/3)/3 + (n+5)*floor(n/2)/8 + floor((n+1)/4)/4. - Vaclav Kotesovec, Aug 18 2015
a(n) = a(n-4) + A001399(n). - Ece Uslu, Esin Becenen, Jan 11 2016, corrected Sep 25 2020
a(6*n) - a(6*n+1) - a(6*n+4) + a(6*n+5) = n+1. - Richard Turk, Apr 19 2016
a(n) = a(n-1) + A005044(n+3) for n>0, i.e., first differences is A005044. - Yuchun Ji, Oct 12 2020
From Vladimír Modrák and Zuzana Soltysova, Dec 09 2020: (Start)
a(n) = round((n + 3)^2/12) + Sum_{i=0..floor(n/4)} round((n - 4*i - 1)^2/12).
a(n) = floor(((n + 3)^2 + 4)/12) + Sum_{i=0..floor(n/4)} floor(((n - 4*i - 1)^2 + 4)/12). (End)
a(n) - a(n-3) = A008642(n). - R. J. Mathar, Jun 23 2021
a(n) - a(n-2) = A025767(n). - R. J. Mathar, Jun 23 2021
a(n) = round((2*n^3 + 30*n^2 + 135*n + 175)/288 + (-1)^n*(n+5)/32). - Dave Neary, Oct 28 2021
From Vladimír Modrák, Jul 13 2022: (Start)
a(n) = Sum_{j=0..floor(n/4)} Sum_{i=0..floor(n/3)} ceiling((max(0,n + 1 - 3*i - 4*j))/2).
a(n) = Sum_{i=0..floor(n/4)} floor(((n + 3 - 4*i)^2 + 4)/12). (End)
a(n) = floor(((n+4)^2*(n+7) - 9*(n+4)*(n mod 2) + 32)/144). - Vladimír Modrák, Mar 23 2025

A026820 Euler's table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <= k for 1 <= k <= n. Also number of partitions of n into at most k parts.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 11, 13, 14, 15, 1, 5, 10, 15, 18, 20, 21, 22, 1, 5, 12, 18, 23, 26, 28, 29, 30, 1, 6, 14, 23, 30, 35, 38, 40, 41, 42, 1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56, 1, 7, 19, 34, 47, 58, 65, 70, 73, 75, 76, 77

Offset: 1

Clark Kimberling

			Triangle starts:
  1;
  1, 2;
  1, 2,  3;
  1, 3,  4,  5;
  1, 3,  5,  6,  7;
  1, 4,  7,  9, 10, 11;
  1, 4,  8, 11, 13, 14, 15;
  1, 5, 10, 15, 18, 20, 21, 22;
  1, 5, 12, 18, 23, 26, 28, 29, 30;
  1, 6, 14, 23, 30, 35, 38, 40, 41, 42;
  1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56;
  ...

G. Chrystal, Algebra, Vol. II, p. 558.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.

Alois P. Heinz, Robert G. Wilson v, Rows n = 1..141, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 831. [scanned copy]
Carolyn Echter, Georg Maier, Juan-Diego Urbina, Caio Lewenkopf, and Klaus Richter, Many-body density of states of bosonic and fermionic gases: a combinatorial approach, arXiv:2409.08696 [cond-mat.quant-gas], 2024. See p. 10.
L. Euler, Introductio in Analysin Infinitorum, Book I, chapter XVI.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers.
R. Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, Univ. Wien, 2013.
Sergei Viznyuk, C program.
Sergei Viznyuk, Local copy of C program.
Eric Weisstein's World of Mathematics, Partition Function q.
Index entries for sequences related to partitions - _Reinhard Zumkeller_, Jan 21 2010

Partial sums of rows of A008284, row sums give A058397, central terms give A171985, mirror is A058400.
T(n,n) = A000041(n), T(n,1) = A000012(n), T(n,2) = A008619(n) for n>1, T(n,3) = A001399(n) for n>2, T(n,4) = A001400(n) for n>3, T(n,5) = A001401(n) for n>4, T(n,6) = A001402(n) for n>5, T(n,7) = A008636(n) for n>6, T(n,8) = A008637(n) for n>7, T(n,9) = A008638(n) for n>8, T(n,10) = A008639(n) for n>9, T(n,11) = A008640(n) for n>10, T(n,12) = A008641(n) for n>11, T(n,n-2) = A007042(n-1) for n>2, T(n,n-1) = A000065(n) for n>1.
Cf. A008284, A026840, A134737, A173519.

import Data.List (inits)
a026820 n k = a026820_tabl !! (n-1) !! (k-1)
a026820_row n = a026820_tabl !! (n-1)
a026820_tabl = zipWith
   (\x -> map (p x) . tail . inits) [1..] $ tail $ inits [1..] where
   p 0 _ = 1
   p _ [] = 0
   p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
-- Reinhard Zumkeller, Dec 18 2013

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Apr 21 2012

t[n_, k_] := Length@ IntegerPartitions[n, k]; Table[ t[n, k], {n, 12}, {k, n}] // Flatten
(* Second program: *)
T[n_, k_] := T[n, k] = If[n==0 || k==1, 1, T[n, k-1] + If[k>n, 0, T[n-k, k]]]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)

T(n,k)=my(s); forpart(v=n,s++,,k); s \\ Charles R Greathouse IV, Feb 27 2018

from sage.combinat.partition import number_of_partitions_length
from itertools import accumulate
for n in (1..11):
    print(list(accumulate([number_of_partitions_length(n, k) for k in (1..n)])))
# Peter Luschny, Jul 28 2022

T(T(n,n),n) = A134737(n). - Reinhard Zumkeller, Nov 07 2007
T(A000217(n),n) = A173519(n). - Reinhard Zumkeller, Feb 20 2010
T(n,k) = T(n,k-1) + T(n-k,k). - Thomas Dybdahl Ahle, Jun 13 2011
T(n,k) = Sum_{i=1..min(k,floor(n/2))} T(n-i,i) + Sum_{j=1+floor(n/2)..k} A000041(n-j). - Bob Selcoe, Aug 22 2014 [corrected by Álvar Ibeas, Mar 15 2018]
O.g.f.: Product_{i>=0} 1/(1-y*x^i). - Geoffrey Critzer, Mar 11 2012
T(n,k) = A008284(n+k,k). - Álvar Ibeas, Jan 06 2015

A033485 a(n) = a(n-1) + a(floor(n/2)), a(1) = 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 83, 101, 119, 142, 165, 195, 225, 262, 299, 346, 393, 450, 507, 577, 647, 730, 813, 914, 1015, 1134, 1253, 1395, 1537, 1702, 1867, 2062, 2257, 2482, 2707, 2969, 3231, 3530, 3829, 4175, 4521, 4914, 5307, 5757

Offset: 1

N. J. A. Sloane. This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Philippe Deléham. Entry revised by N. J. A. Sloane, Jun 10 2012

Sequence gives the number of partitions of 2n into "strongly decreasing" parts (see the function s*(n) in the paper by Bessenrodt, Olsson, and Sellers); see the example in A040039.
a(A036554(n)) is even, a(A003159(n)) is odd. - Benoit Cloitre, Oct 23 2002
Partial sums of the sequence a(1)=1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), ...; example: a(1) = 1, a(2) = 1+1 = 2, a(3) = 1+1+1 = 3, a(4) = 1+1+1+2 = 5, a(5) = 1+1+1+2+2 = 7, ... - Philippe Deléham, Jan 02 2004
The number of odd numbers before the n-th even number in this sequence is A003156(n). - Philippe Deléham, Mar 27 2004
There are no terms divisible by 4 and there are infinitely many terms divisible by {2,3,5,6,7,9,10,11,13,14,15} (see van Doorn link). - Ivan N. Ianakiev, Aug 06 2022 and Wouter van Doorn, Sep 17 2024
a(n) = A001401(n), for 1..14. A001401(15) = 84. - Wolfdieter Lang, Jan 09 2023

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
J. Arkin, Problem H-102: Gone but not forgotten, Fibonacci Quarterly, Vol. 9 (1971), page 135.
Christine Bessenrodt, Jorn B. Olsson, and James A. Sellers, Unique path partitions: Characterization and Congruences, arXiv:1107.1156 [math.CO], 2011-2012.
Philippe Deléham, Letter to N. J. A. Sloane, Apr 20 1998
William Gasarch, What is known about that sequence, Computational Complexity blog, Jul 20 2022.
Wouter van Doorn, On the congruence properties and growth rate of a recursively defined sequence, arXiv:2406.09532 [math.NT], 2024.

Cf. A040039 (first differences), A178855 (partial sums).
Also half of A000123 (with first term omitted).
Cf. A022907.
Cf. A036554, A003159, A003156.
Cf. A007413, A178855, A131205.

import Data.List (transpose)
a033485 n = a033485_list !! (n-1)
a033485_list = 1 : zipWith (+)
   a033485_list (concat $ transpose [a033485_list, a033485_list])
-- Reinhard Zumkeller, Nov 15 2012

[n le 1 select 1 else Self(n-1) + Self(Floor(n/2)) : n in [1..60]]; // Vincenzo Librandi, Nov 20 2015

a:= proc(n) option remember;
      `if`(n<2, n, a(n-1)+a(iquo(n, 2)))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 16 2019

b[1]=1; b[n_] := b[n]=Sum[b[k], {k, 1, n/2}]; Table[b[n], {n, 3, 105, 2}] (* Robert G. Wilson v, Apr 22 2001 *)

PARI
```
a(n)=if(n<2,1,a(floor(n/2))+a(n-1))
```

from itertools import islice
from collections import deque
def A033485_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 2)
    while True:
        a += b
        yield a
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A033485_list = list(islice(A033485_gen(),40)) # Chai Wah Wu, Jun 07 2022

a(n) = A000123(n)/2, for n >= 1.
Conjecture: lim_{n->oo} a(2n)/a(n)*log(n)/n = c = 1.64.... and a(n)/A(n) is bounded where A(n)=1 if n is a power of 2, otherwise A(n) = sqrt(n)*Product_{kBenoit Cloitre, Jan 26 2003
G.f.: A(x) satisfies x + (1+x)*A(x^2) = (1-x)*A(x). a(n) modulo 2 = A035263(n). - Philippe Deléham, Feb 25 2004
G.f.: (1/2)*(((1-x)*Product_{n>=0} (1-x^(2^n)))^(-1)-1). a(n) modulo 4 = A007413(n). - Philippe Deléham, Feb 28 2004
Sum_{k=1..n} a(k) = (a(2n+1)-1)/2 = A178855(n). - Philippe Deléham, Mar 18 2004
a(2n-1) = A131205(n). - Jean-Paul Allouche, Aug 11 2021
There exists a function f(n) such that n^f(n) < a(n) < n^(f(n) + epsilon) for all epsilon > 0 and all large enough n. - Wouter van Doorn, Sep 17 2024

A026811 Number of partitions of n in which the greatest part is 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 84, 101, 119, 141, 164, 192, 221, 255, 291, 333, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 1342, 1469, 1602, 1747, 1898, 2062, 2233, 2418, 2611, 2818, 3034, 3266, 3507, 3765

Offset: 0

Clark Kimberling

Essentially same as A001401: five zeros followed by A001401.

Also number of partitions of n into exactly 5 parts.

D. E. Knuth, The Art of Computer Programming, vol. 4, fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

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

Cf. A026810, A026812, A026813, A026814, A026815, A026816, A002622 (partial sums), A008667 (first differences).

List([0..70],n->NrPartitions(n,5)); # Muniru A Asiru, May 17 2018

Table[Count[IntegerPartitions[n], {5, _}], {n, 0, 55}] (* corrected by Harvey P. Dale, Oct 24 2011 *)
Table[Length[IntegerPartitions[n, {5}]], {n, 0, 55}] (* Eric Rowland, Mar 02 2017 *)
CoefficientList[Series[x^5/Product[1 - x^k, {k, 1, 5}], {x, 0, 65}], x] (* Robert A. Russell, May 13 2018 *)
Drop[LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1}, Append[Table[0,{14}],1],110],9] (* Robert A. Russell, May 17 2018 *)

a(n)=round((n^4+10*(n^3+n^2)-75*n-45*(-1)^n*n)/2880);
for(n=0,10000,print(n," ",a(n))); /* b-file format */
/* Washington Bomfim, Jul 03 2012 */

x='x+O('x^99); concat(vector(5), Vec(x^5/prod(k=1, 5, 1-x^k))) \\ Altug Alkan, May 17 2018

a(n) = round( ((n^4+10*(n^3+n^2)-75*n -45*n*(-1)^n)) / 2880 ). - Washington Bomfim, Jul 03 2012
G.f.: x^5/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)). - Joerg Arndt, Jul 04 2012
a(n) = A008284(n,5). - Robert A. Russell, May 13 2018
From Gregory L. Simay, Jul 28 2019: (Start)
a(2n)   = a(2n-1) + a(n+1) + a(n) - a(n-3) - a(n-4);
a(2n+1) = a(2n)   + a(n+3) - a(n-5). (End)
From R. J. Mathar, Jun 23 2021: (Start)
a(n) - a(n-5) = A001400(n-5).
a(n) - a(n-4) = A008669(n-5).
a(n) - a(n-3) = A029007(n-5).
a(n) - a(n-2) = A029032(n-5).
a(n) = +a(n-1) +a(n-2) -a(n-5) -a(n-6) -a(n-7) +a(n-8) +a(n-9) +a(n-10) -a(n-13) -a(n-14) +a(n-15). (End)

More terms from Robert G. Wilson v, Jan 11 2002

a(0)=0 inserted by Joerg Arndt, Jul 04 2012

A001402 Number of partitions of n into at most 6 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 8442, 9192

Offset: 0

N. J. A. Sloane

Also number of partitions of n into parts <= 6: a(n) = A026820(n,6). - Reinhard Zumkeller, Jan 21 2010
Counts unordered closed walks of weight n on a single vertex graph containing 6 loops of weights 1, 2, 3, 4, 5 and 6. - David Neil McGrath, Apr 11 2015
Number of different distributions of n+21 identical balls in 6 boxes as x,y,z,p,q,m where 0Ece Uslu and Esin Becenen, Jan 11 2016
a(n) could be the total number of non-isomorphic geodetic graphs of diameter n>=2 homeomorphic to the Petersen graph. - Carlos Enrique Frasser, May 24 2018

			The number of partitions of 6 into parts less than or equal to 6 is a(6)=11. These are (6)(51)(42)(33)(411)(321)(222)(3111)(2211)(21111)(111111). - _David Neil McGrath_, Apr 11 2015
a(4) = 5, i.e., {1,2,3,4,5,10},{1,2,3,4,6,9},{1,2,3,4,7,8},{1,2,3,5,6,8},{1,2,4,5,6,7} Number of different distributions of 25 identical balls in 6 boxes as x,y,z,p,q,m where 0 < x < y < z < p < q < m. - _Ece Uslu_, Esin Becenen, Jan 11 2016

A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 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).

Washington Bomfim, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
G. E. Andrews, Partitions: At the Interface of q-Series and Modular Forms, The Ramanujan Journal 7, 385-400 (2003), Eq.(3.10).
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, American Journal of Mathematics, Vol. 2, No. 1 (Mar., 1879), pp. 71-84 (14 pages).
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. [Annotated scanned copy]
C. E. Frasser and G. N. Vostrov, Geodetic Graphs Homeomorphic to a Given Geodetic Graph, arXiv:1611.01873 [cs.DM], 2016. [p. 27]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 355
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], 2011.
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Essentially same as A026812. Cf. A037145 (first differences), A288341 (partial sums).
a(n) = A008284(n+6, 6), n >= 0.
A194197(n) = a(60*n). - Alois P. Heinz, Aug 23 2011

with(combstruct):ZL7:=[S,{S=Set(Cycle(Z,card<7))}, unlabeled]: seq(count(ZL7,size=n),n=0..50);  # Zerinvary Lajos, Sep 24 2007
a:= n-> (Matrix(21, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 1, 0, 0, -1, 0, -2, 0, 1, 1, 1, 1, 0, -2, 0, -1, 0, 0, 1, 1, -1][i] else 0 fi)^n)[1,1]; seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2008
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=6)},unlabelled]: seq(combstruct[count](B, size=n), n=0..50); # Zerinvary Lajos, Mar 21 2009
## more efficient for large arguments (try with 10^100 or 100^1000):
a:= proc(n) local m, r; m := iquo (n, 60, 'r');
(167 +(2325 +(15400 +(47250 +54000*m +4500*r)*m +3150*r +150*r^2)*m
+[0, 795, 1875, 3030, 4500, 6075, 7995, 10050, 12480, 15075, 18075, 21270, 24900, 28755, 33075, 37650, 42720, 48075, 53955, 60150, 66900, 73995, 81675, 89730, 98400, 107475, 117195, 127350, 138180, 149475, 161475, 173970, 187200, 200955, 215475, 230550, 246420, 262875, 280155, 298050, 316800, 336195, 356475, 377430, 399300, 421875, 445395, 469650, 494880, 520875, 547875, 575670, 604500, 634155, 664875, 696450, 729120, 762675, 797355, 832950][r+1])*m
+[0, 63, 207, 348, 570, 795, 1143, 1482, 1968, 2475, 3135, 3828, 4722, 5643, 6795, 8010, 9468, 11007, 12843, 14760, 17010, 19383, 22107, 24978, 28260, 31695, 35583, 39672, 44238, 49035, 54375, 59958, 66132, 72603, 79695, 87120, 95238, 103707, 112923, 122550, 132960, 143823, 155547, 167748, 180870, 194535, 209163, 224382, 240648, 257535, 275535, 294228, 314082, 334683, 356535, 379170, 403128, 427947, 454143, 481260][r+1])*m/6
+[1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 8442, 9192, 9975, 10829, 11720, 12692, 13702, 14800, 15944, 17180, 18467][r+1] end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 22 2011
A := [1,1,2,3,5,7,11,14,20,26,35,44,58,71,90,110,136,163,199,235,282];
a := proc(n) option remember; if n < 21 then A[n+1] else 1+(a(n-2)+a(n-3)+a(n-4))-(2*a(n-7)+2*a(n-8)+a(n-9))+(a(n-11)+2*a(n-12)+2*a(n-13))-(a(n-16)+a(n-17)+a(n-18))+(a(n-20)) fi end:
seq(a(i),i=0..50); # Peter Luschny, Aug 23 2011
## program using quasi-polynomials; see article by Sills and Zeilberger:
a:= m-> subs (n=m, add ([[n^5/86400 +7*n^4/11520 +77*n^3/6480 +245*n^2/2304 +43981*n/103680 +199577/345600], [-n^2/768 -7*n/256 -581/4608, n^2/768 +7*n/256 +581/4608], [-n/162 -19/324, -n/162 -23/324, n/81 +7/54], [1/32, -1/32, -1/32, 1/32], [1/25, 0, -1/25, -2/25, 2/25], [1/36, -1/36, -1/18, -1/36, 1/36, 1/18]][r][1 +irem (m-1+r, r)], r=1..6)):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 24 2011
## using Andrews-style expressions; see article by Sills and Zeilberger:
a:= n-> 1 +31*n^2/288 +floor(n/4)/16 -floor(n/4 +1/2)/16 +7*n^4/11520 +floor(n/5)/5 +n^5/86400 -(n^2/384 +7*n/128 +581/2304)*n +(n^2/192 +7*n/64 +581/1152) *floor(n/2) -(n/54 +61/324)*n +(n/54 +19/108) *floor((n+1)/3) +(n/27 +7/18) *floor(n/3) +floor(n/6)/18 -floor(n/6 +2/3)/36 +floor(n/6 +1/3)/18 +floor((n+1)/6)/12 +713*n/1800 +77*n^3/6480:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 24 2011

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)*(1 - x^5)*(1 - x^6)), {x, 0, 60} ], x ]
(* Second program: *)
T[n_, k_] := T[n, k] = If[n==0 || k==1, 1, T[n, k-1] + If[k>n, 0, T[n-k, k]]]; a[n_] := T[n, 6]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 12 2017, after Alois P. Heinz's code for A026820 *)
Table[Length[IntegerPartitions[n,6]],{n,0,50}] (* Harvey P. Dale, Jul 30 2025 *)

a(n)=floor((6*n^5+315*n^4+6160*n^3+55125*n^2+(216705+9600*(n%3<1))*n+527500)/518400+(n+1)*(n+20)*(-1)^n/768) \\ Tani Akinari, May 27 2014

a(n)={round((n+11)*((6*n^4+249*n^3+2071*n^2-4931*n+40621)/518400+n\2*(n+10) /192+( (n+1)\3+ n\3*2 )/54))};
vector(60,n,n--; a(n)) \\ Washington Bomfim, Jan 16 2021

a(n) = 1 + (a(n-2) + a(n-3) + a(n-4)) - (2*a(n-7) + 2*a(n-8) + a(n-9)) + (a(n-11) + 2*a(n-12) + 2*a(n-13)) - (a(n-16) + a(n-17) + a(n-18)) + (a(n-20)). - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)). - Alois P. Heinz, Aug 22 2011
a(n) ~ n^5 / 86400. - Charles R Greathouse IV, Aug 23 2011
a(n) = (167 + (2325 + (15400 + (47250 + 54000*m + 4500*r)*m + 3150*r + 150*r^2)*m + X(r))*m + Y(r))*m/6 + Z(r) where m = floor(n/60), r = n mod 60 and X, Y, Z are functions of r (see Maple program below). - Alois P. Heinz, Aug 23 2011
a(n) = floor((2 + 3*(floor(n/3) + floor(-n/3))) * (floor(n/3)+1)/54 + (6*n^5 + 315*n^4 + 6160*n^3 + 55125*n^2 + 219905*n + 485700)/518400 + (n+1)*(n+20)*(-1)^n/768). - Tani Akinari, Aug 05 2013
a(n) = a(n-1) + a(n-2) - a(n-5) - 2*a(n-7) + a(n-9) + a(n-10) + a(n-11) + a(n-12) - 2*a(n-14) - a(n-16) + a(n-19) + a(n-20) - a(n-21). - David Neil McGrath, Apr 11 2015
a(n+6) = a(n) + A001401(n). - Ece Uslu, Esin Becenen, Jan 11 2016
a(n) = round((n+11)*((6*n^4 + 249*n^3 + 2071*n^2 - 4931*n + 40621)/518400 + floor(n/2)*(n+10)/192 + (floor((n+1)/3) + 2*floor(n/3))/54)). - Washington Bomfim, Jan 15 2021

cp's OEIS Frontend

A008619 Positive integers repeated.

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A001399 a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.

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A001400 Number of partitions of n into at most 4 parts.

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A026820 Euler's table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <= k for 1 <= k <= n. Also number of partitions of n into at most k parts.

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A033485 a(n) = a(n-1) + a(floor(n/2)), a(1) = 1.

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A008667 Expansion of g.f.: 1/((1-x^2)(1-x^3)(1-x^4)*(1-x^5)).