A003105 -id:A003105 - cp's OEIS frontend

A002865 Number of partitions of n that do not contain 1 as a part.

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701

Offset: 0

N. J. A. Sloane

Also the number of partitions of n-1, n >= 2, such that the least part occurs exactly once. See A096373, A097091, A097092, A097093. - Robert G. Wilson v, Jul 24 2004 [Corrected by Wolfdieter Lang, Feb 18 2009]
Number of partitions of n+1 where the number of parts is itself a part. Take a partition of n (with k parts) which does not contain 1, remove 1 from each part and add a new part of size k+1. - Franklin T. Adams-Watters, May 01 2006
Number of partitions where the largest part occurs at least twice. - Joerg Arndt, Apr 17 2011
Row sums of triangle A147768. - Gary W. Adamson, Nov 11 2008
From Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start)
a(n) is the number of sets of n disjoint pairs of 2n things, called a pairing, disjoint with a given pairing (A053871), that are unique under permutations preserving the given pairing.
Can be seen immediately from a graphical representation which must decompose into even numbered cycles of 4 or more things, as connected by pairs alternating between the pairings. Each thing is in a single cycle, so this is a partition of 2n into even parts greater than 2, equivalent to a partition of n into parts greater than 1. (End)
Convolution product (1, 1, 2, 2, 4, 4, ...) * (1, 2, 3, ...) = A058682 starting (1, 3, 7, 13, 23, 37, ...); with row sums of triangle A171239 = A058682. - Gary W. Adamson, Dec 05 2009
Also the number of 2-regular multigraphs with loops forbidden. - Jason Kimberley, Jan 05 2011
Number of appearances of the multiplicity n, n-1, ..., n-k in all partitions of n, for k < n/2. (Only populated by multiplicities of large numbers of 1's.) - William Keith, Nov 20 2011
Also the number of equivalence classes of n X n binary matrices with exactly 2 1's in each row and column, up to permutations of rows and columns (cf. A133687). - N. J. A. Sloane, Sep 16 2013
The q-Catalan numbers ((1-q)/(1-q^(n+1)))[2n,n]q, where [2n,n]_q are the central q-binomial coefficients, match this sequence in their initial segment of length n. - _William J. Keith, Nov 14 2013
Starting at a(2) this sequence gives the number of vertices on a nim tree created in the game of edge removal for a path P_{n} where n is the number of vertices on the path. This is the number of nonisomorphic graphs that can result from the path when the game of edge removal is played. - Lyndsey Wong, Jul 09 2016
The number of different ways to climb a staircase taking at least two stairs at a time. - Mohammad K. Azarian, Nov 20 2016
Let 1,0,1,1,1,... (offset 0) count unlabeled, connected, loopless 1-regular digraphs. This here is the Euler transform of that sequence, counting unlabeled loopless 1-regular digraphs. A145574 is the associated multiset transformation. A000166 are the labeled loopless 1-regular digraphs. - R. J. Mathar, Mar 25 2019
For n > 1, also the number of partitions with no part greater than the number of ones. - George Beck, May 09 2019 [See A187219 which is the correct sequence for this interpretation for n >= 1. - Spencer Miller, Jan 30 2023]
From Gus Wiseman, May 19 2019: (Start)
Conjecture: Also the number of integer partitions of n - 1 that have a consecutive subsequence summing to each positive integer from 1 to n - 1. For example, (32211) is such a partition because we have consecutive subsequences:
  1: (1)
  2: (2)
  3: (3) or (21)
  4: (22) or (211)
  5: (32) or (221)
  6: (2211)
  7: (322)
  8: (3221)
  9: (32211)
(End)
There is a sufficient and necessary condition to characterize the partitions defined by Gus Wiseman. It is that the largest part must be less than or equal to the number of ones plus one. Hence, the number of partitions of n with no part greater than the number of ones is the same as the number of partitions of n-1 that have a consecutive subsequence summing to each integer from 1 to n-1. Gus Wiseman's conjecture can be proved bijectively. - Andrew Yezhou Wang, Dec 14 2019
From Peter Bala, Dec 01 2024: (Start)
Let P(2, n) denote the set of partitions of n into parts k > 1. Then A000041(n) = - Sum_{parts k in all partitions in P(2, n+2)} mu(k). For example, with n = 5, there are 4 partitions of n + 2 = 7 into parts greater than 1, namely, 7, 5 + 2, 4 + 3, 3 + 2 + 2, and mu(7) + (mu(5) + mu(2)) + (mu(4 ) + mu(3)) + (mu(3) + mu(2) + mu(2)) = -7 = - A000041(5). (End)

			a(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2.
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 7*x^8 + 8*x^9 + ...
From _Gus Wiseman_, May 19 2019: (Start)
The a(2) = 1 through a(9) = 8 partitions not containing 1 are the following. The Heinz numbers of these partitions are given by A005408.
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (22)  (32)  (33)   (43)   (44)    (54)
                        (42)   (52)   (53)    (63)
                        (222)  (322)  (62)    (72)
                                      (332)   (333)
                                      (422)   (432)
                                      (2222)  (522)
                                              (3222)
The a(2) = 1 through a(9) = 8 partitions of n - 1 whose least part appears exactly once are the following. The Heinz numbers of these partitions are given by A247180.
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (21)  (31)  (32)   (42)   (43)    (53)
                        (41)   (51)   (52)    (62)
                        (221)  (321)  (61)    (71)
                                      (331)   (332)
                                      (421)   (431)
                                      (2221)  (521)
                                              (3221)
The a(2) = 1 through a(9) = 8 partitions of n + 1 where the number of parts is itself a part are the following. The Heinz numbers of these partitions are given by A325761.
  (21)  (22)  (32)   (42)   (52)    (62)    (72)     (82)
              (311)  (321)  (322)   (332)   (333)    (433)
                            (331)   (431)   (432)    (532)
                            (4111)  (4211)  (531)    (631)
                                            (4221)   (4222)
                                            (4311)   (4321)
                                            (51111)  (4411)
                                                     (52111)
The a(2) = 1 through a(8) = 7 partitions of n whose greatest part appears at least twice are the following. The Heinz numbers of these partitions are given by A070003.
  (11)  (111)  (22)    (221)    (33)      (331)      (44)
               (1111)  (11111)  (222)     (2221)     (332)
                                (2211)    (22111)    (2222)
                                (111111)  (1111111)  (3311)
                                                     (22211)
                                                     (221111)
                                                     (11111111)
Nonisomorphic representatives of the a(2) = 1 through a(6) = 4 2-regular multigraphs with n edges and n vertices are the following.
  {12,12}  {12,13,23}  {12,12,34,34}  {12,12,34,35,45}  {12,12,34,34,56,56}
                       {12,13,24,34}  {12,13,24,35,45}  {12,12,34,35,46,56}
                                                        {12,13,23,45,46,56}
                                                        {12,13,24,35,46,56}
The a(2) = 1 through a(9) = 8 partitions of n with no part greater than the number of ones are the following. The Heinz numbers of these partitions are given by A325762.
  (11)  (111)  (211)   (2111)   (2211)    (22111)    (22211)     (33111)
               (1111)  (11111)  (3111)    (31111)    (32111)     (222111)
                                (21111)   (211111)   (41111)     (321111)
                                (111111)  (1111111)  (221111)    (411111)
                                                     (311111)    (2211111)
                                                     (2111111)   (3111111)
                                                     (11111111)  (21111111)
                                                                 (111111111)
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, p*(n).
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
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).
P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 334.

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. P. Akande et al., Computational study of non-unitary partitions, arXiv:2112.03264 [math.CO], 2021.
Colin Albert, Olivia Beckwith, Irfan Demetoglu, Robert Dicks, John H. Smith, and Jasmine Wang, Integer partitions with large Dyson rank, arXiv:2203.08987 [math.NT], 2022.
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, (2024). See p. 6.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
G. Dahl and T. A. Haufmann, Zero-one completely positive matrices and the A(R,S) classes, Preprint, 2016.
Atul Dixit, Gaurav Kumar, and Aviral Srivastava, Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions, arXiv:2508.04359 [math.CO], 2025. See references.
R. P. Gallant, G. Gunther, B. L. Hartnell, and D. F. Rall, A game of edge removal on graphs, JCMCC, 57 (2006), 75 - 82.
Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
H. Gropp, On tactical configurations, regular bipartite graphs and (v,k,even)-designs, Discr. Math., 155 (1996), 81-98.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 100
Wenwei Li, Approximation of the Partition Number After Hardy and Ramanujan: An Application of Data Fitting Method in Combinatorics, arXiv preprint arXiv:1612.05526 [math.NT], 2016-2018.
Wenwei Li, On the Number of Conjugate Classes of Derangements, arXiv:1612.08186 [math.CO], 2016.
J. L. Nicolas and A. Sárközy, On partitions without small parts, Journal de théorie des nombres de Bordeaux, 12 no. 1 (2000), p. 227-254.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
H. P. Robinson, Letter to N. J. A. Sloane, Jul 12 1971
H. P. Robinson, Letter to N. J. A. Sloane, Dec 10 1973
H. P. Robinson, Letter to N. J. A. Sloane, Jan 4 1974.
Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares, arXiv:2103.11018 [cs.DM], 2021.
Miloslav Znojil, Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities, arXiv:2010.15014 [quant-ph], 2020.
Miloslav Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, arXiv:2102.12272 [quant-ph], 2021.
Miloslav Znojil, Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks, arXiv:2108.07110 [quant-ph], 2021.
Index entries for related partition-counting sequences

First differences of partition numbers A000041. Cf. A053445, A072380, A081094, A081095, A232697.
Pairwise sums seem to be in A027336.
Essentially the same as A085811.
Cf. A025147, A147768, A058682, A171239.
A column of A090824 and of A133687 and of A292508 and of A292622. Cf. A229161.
2-regular not necessarily connected graphs: A008483 (simple graphs), A000041 (multigraphs with loops allowed), this sequence (multigraphs with loops forbidden), A027336 (graphs with loops allowed but no multiple edges). - Jason Kimberley, Jan 05 2011
See also A098743 (parts that do not divide n).
Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy: A274161. - Lyndsey Wong, Jul 09 2016
Cf. A002033, A070003, A103295, A247180, A325676, A325684, A325761, A325762, A325763.
Row sums of characteristic array A145573.
Number of partitions of n into parts >= m: A008483 (m = 3), A008484 (m = 4), A185325 - A185329 (m = 5 through 9).

Concatenation([1],List([1..41],n->NrPartitions(n)-NrPartitions(n-1))); # Muniru A Asiru, Aug 20 2018

A41 := func; [A41(n)-A41(n-1):n in [0..50]]; // Jason Kimberley, Jan 05 2011

with(combstruct): ZL1:=[S, {S=Set(Cycle(Z,card>1))}, unlabeled]: seq(count(ZL1,size=n), n=0..50);  # Zerinvary Lajos, Sep 24 2007
G:= {P=Set (Set (Atom, card>1))}: combstruct[gfsolve](G, unlabeled, x): seq  (combstruct[count] ([P, G, unlabeled], size=i), i=0..50);  # Zerinvary Lajos, Dec 16 2007
with(combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, unlabeled]; end: A:=a(2):seq(count(A, size=n), n=0..50);  # Zerinvary Lajos, Jun 11 2008
# alternative Maple program:
A002865:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)-1)*A002865(n-j), j=1..n)/n)
    end:
seq(A002865(n), n=0..60);  # Alois P. Heinz, Sep 17 2017

Table[ PartitionsP[n + 1] - PartitionsP[n], {n, -1, 50}] (* Robert G. Wilson v, Jul 24 2004 *)
f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 2], {n, 50}] (* Robert G. Wilson v *)
Table[SeriesCoefficient[Exp[Sum[x^(2*k)/(k*(1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Aug 18 2018 *)
CoefficientList[Series[1/QPochhammer[x^2, x], {x,0,50}], x] (* G. C. Greubel, Nov 03 2019 *)
Table[Count[IntegerPartitions[n],?(FreeQ[#,1]&)],{n,0,50}] (* _Harvey P. Dale, Feb 12 2023 *)

{a(n) = if( n<0, 0, polcoeff( (1 - x) / eta(x + x * O(x^n)), n))};

a(n)=if(n,numbpart(n)-numbpart(n-1),1) \\ Charles R Greathouse IV, Nov 26 2012

from sympy import npartitions
def A002865(n): return npartitions(n)-npartitions(n-1) if n else 1 # Chai Wah Wu, Mar 30 2023

def A002865_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+2)) for m in (0..60)) ).list()
A002865_list(50) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>1} 1/(1-x^m).
a(0)=1, a(n) = p(n) - p(n-1), n >= 1, with the partition numbers p(n) := A000041(n).
a(n) = A085811(n+3). - James Sellers, Dec 06 2005 [Corrected by Gionata Neri, Jun 14 2015]
a(n) = A116449(n) + A116450(n). - Reinhard Zumkeller, Feb 16 2006
a(n) = Sum_{k=2..floor((n+2)/2)} A008284(n-k+1,k-1) for n > 0. - Reinhard Zumkeller, Nov 04 2007
G.f.: 1 + Sum_{n>=2} x^n / Product_{k>=n} (1 - x^k). - Joerg Arndt, Apr 13 2011
G.f.: Sum_{n>=0} x^(2*n) / Product_{k=1..n} (1 - x^k). - Joerg Arndt, Apr 17 2011
a(n) = A090824(n,1) for n > 0. - Reinhard Zumkeller, Oct 10 2012
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - Vaclav Kotesovec, Feb 26 2015, extended Nov 04 2016
G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(0) = 1, a(n) = A232697(n) - 1. - George Beck, May 09 2019
From Peter Bala, Feb 19 2021: (Start)
G.f.: A(q) = Sum_{n >= 0} q^(n^2)/( (1 - q)*Product_{k = 2..n} (1 - q^k)^2 ).
More generally, A(q) = Sum_{n >= 0} q^(n*(n+r))/( (1 - q) * Product_{k = 2..n} (1 - q^k)^2 * Product_{i = 1..r} (1 - q^(n+i)) ) for r = 0,1,2,.... (End)
G.f.: 1 + Sum_{n >= 1} x^(n+1)/Product_{k = 1..n-1} 1 - x^(k+2). - Peter Bala, Dec 01 2024

A001651 Numbers not divisible by 3.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104

Offset: 1

N. J. A. Sloane

Inverse binomial transform of A084858. - Benoit Cloitre, Jun 12 2003
Earliest monotonic sequence starting with (1,2) and satisfying the condition: "a(n)+a(n-1) is not in the sequence." - Benoit Cloitre, Mar 25 2004. [The numbers of the form a(n)+a(n-1) form precisely the complement with respect to the positive integers. - David W. Wilson, Feb 18 2012]
a(1) = 1; a(n) is least number which is relatively prime to the sum of all the previous terms. - Amarnath Murthy, Jun 18 2001
For n > 3, numbers having 3 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005
Also numbers n such that (n+1)*(n+2)/6 = A000292(n)/n is an integer. - Ctibor O. Zizka, Oct 15 2010
Notice the property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 3). - Bruno Berselli, Nov 17 2010
A001651 mod 9 gives A141425. - Paul Curtz, Dec 31 2010. (Correct for the modified offset 1. - M. F. Hasler, Apr 07 2015)
The set of natural numbers (1, 2, 3, ...), sequence A000027; represents the numbers of ordered compositions of n using terms in the signed set: (1, 2, -4, -5, 7, 8, -10, -11, 13, 14, ...). This follows from (1, 2, 3, ...) being the INVERT transform of A011655, signed and beginning: (1, 1, 0, -1, -1, 0, 1, 1, 0, ...). - Gary W. Adamson, Apr 28 2013
Union of A047239 and A047257. - Wesley Ivan Hurt, Dec 19 2013
Numbers whose sum of digits (and digital root) is != 0 (mod 3). - Joerg Arndt, Aug 29 2014
The number of partitions of 3*(n-1) into at most 2 parts. - Colin Barker, Apr 22 2015
a(n) is the number of partitions of 3*n into two distinct parts. - L. Edson Jeffery, Jan 14 2017
Conjectured (and like even easily proved) to be the graph bandwidth of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017
Numbers k such that Fibonacci(k) mod 4 = 1 or 3. Equivalently, sequence lists the indices of the odd Fibonacci numbers (see A014437). - Bruno Berselli, Oct 17 2017
Minimum value of n_3 such that the "rectangular spiral pattern" is the optimal solution for Ripà's n_1 X n_2 x n_3 Dots Problem, for any n_1 = n_2. For example, if n_1 = n_2 = 5, n_3 = floor((3/2)*(n_1 - 1)) + 1 = a(5). - Marco Ripà, Jul 23 2018
For n >= 54, a(n) = sat(n, P_n), the minimum number of edges in a P_n-saturated graph on n vertices, where P_n is the n-vertex path (see Dudek, Katona, and Wojda, 2003; Frick and Singleton, 2005). - Danny Rorabaugh, Nov 07 2017
From Roger Ford, May 09 2021: (Start)
a(n) is the smallest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.
      /\   The top arch has a length of 3. /\   The top arch has a length of 3.
     /  \  Both bottom arches have a      //\\  The middle arch has a length of 2.
    //\/\\ length of 1.                  ///\\\ The bottom arch has a length of 1.
  Example: a(6) = 8     /\      /\
                       //\\ /\ //\\ /\   2 + 1 + 1 + 2 + 1 + 1 = 8. (End)
This is the lexicographically earliest increasing sequence of positive integers such that no polynomial of degree d can be fitted to d+2 consecutive terms (equivalently, such that no iterated difference is zero). - Pontus von Brömssen, Dec 26 2021

			G.f.: x + 2*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 10*x^7 + 11*x^8 + 13*x^9 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., Vol. 11, No. 4 (1973), pp. 337-386.
Aneta Dudek, Gyula Y. Katona, and A.Pawel Wojda, m_Path Cover Saturated Graphs, Electronic Notes in Discrete Math., Vol. 13 (April 2003), pp. 41-44.
Marietjie Frick and Joy Singleton, Lower Bound for the Size of Maximal Nontraceable Graphs, Electron. J. Combin., 12#R32 (2005), 9 pp.
Aviezri S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)
Brian Hopkins, Euler's Enumerations, Enumerative Combinatorics and Applications, Vol. 1, No. 1 (2021), Article #S1H1.
G. Ledin, Jr., Is Eratosthenes out?, Fib. Quart., Vol. 6, No. 4 (1968), pp. 261-265.
Gerard P. Michon, Counting Polyhedra.
Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, Vol. 120, No. 5 (2013), pp. 409-429 [see p. 417].
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly, Vol. 109, No. 6 (2002), pp. 559-564, Ex. 2.2.
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.
Marco Ripà, The rectangular spiral or the n1 X n2 X ... X nk Points Problem , Notes on Number Theory and Discrete Mathematics, Vol. 20, No. 1 (2014), pp. 59-71.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph, Graph Bandwidth, and RATS Sequence.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Differs from A059564 after 35 = a(24) = A059564(24).
Cf. A000726, A001082, A003105, A005408 (n=1 or 3 mod 4), A007494, A008585 (complement), A011655, A026386, A032766, A073010, A191967, A225227, A004526, A248897.
Cf. A000027, A000217, A000292, A000982, A001477, A008619, A014437, A040001, A047239, A047257, A077043, A084858, A113801, A141425, A215879.

Filtered([0..110],n->n mod 3<>0); # Muniru A Asiru, Jul 24 2018

a001651 = (`div` 2) . (subtract 1) . (* 3)
a001651_list = filter ((/= 0) . (`mod` 3)) [1..]
-- Reinhard Zumkeller, Jul 07 2012, Aug 23 2011

[3*(2*n-1)/4-(-1)^n/4: n in [1..80]]; // Vincenzo Librandi, Jun 07 2011

A001651 := n -> 3*floor(n/2) - (-1)^n; # Corrected by M. F. Hasler, Apr 07 2015
A001651:=(1+z+z**2)/(z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation
a[1]:=1:a[2]:=2:for n from 3 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=1..69); # Zerinvary Lajos, Mar 16 2008, offset corrected by M. F. Hasler, Apr 07 2015

Select[Table[n,{n,200}],Mod[#,3]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
Drop[Range[200 + 1], {1, -1, 3}] - 1 (* József Konczer, May 24 2016 *)
Floor[(3 Range[70] - 1)/2] (* Eric W. Weisstein, Apr 24 2017 *)
CoefficientList[Series[(x^2 + x + 1)/((x - 1)^2 (x + 1)), {x, 0, 70}],
  x] (* or *)
LinearRecurrence[{1, 1, -1}, {1, 2, 4}, 70] (* Robert G. Wilson v, Jul 25 2018 *)

{a(n) = n + (n-1)\2}; /* Michael Somos, Jan 15 2011 */

x='x+O('x^100); Vec(x*(1+x+x^2)/((1-x)*(1-x^2))) \\ Altug Alkan, Oct 22 2015

print([k for k in range(1, 105) if k%3]) # Michael S. Branicky, Sep 06 2021

def A001651(n): return (n<<1)-(n>>1)-1 # Chai Wah Wu, Mar 05 2024

a(n) = 3 + a(n-2) for n > 2.
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
a(2*n+1) = 3*n+1, a(2*n) = 3*n-1.
G.f.: x * (1 + x + x^2) / ((1 - x) * (1 - x^2)). - Michael Somos, Jun 08 2000
a(n) = (4-n)*a(n-1) + 2*a(n-2) + (n-3)*a(n-3) (from the Carlitz et al. article).
a(n) = floor((3*n-1)/2). [Corrected by Gary Detlefs]
a(1) = 1, a(n) = 2*a(n-1) - 3*floor(a(n-1)/3). - Benoit Cloitre, Aug 17 2002
a(n+1) = 1 + n - n mod 2 + (n + n mod 2)/2. - Reinhard Zumkeller, Dec 17 2002
a(1) = 1, a(n+1) = a(n) + (a(n) mod 3). - Reinhard Zumkeller, Mar 23 2003
a(1) = 1, a(n) = 3*(n-1) - a(n-1). - Benoit Cloitre, Apr 12 2003
a(n) = 3*(2*n-1)/4 - (-1)^n/4. - Benoit Cloitre, Jun 12 2003
Nearest integer to (Sum_{k>=n} 1/k^3)/(Sum_{k>=n} 1/k^4). - Benoit Cloitre, Jun 12 2003
Partial sums of A040001. a(n) = A032766(n-1)+1. - Paul Barry, Sep 02 2003
a(n) = T(n, 1) = T(n, n-1), where T is the array in A026386. - Emeric Deutsch, Feb 18 2004
a(n) = sqrt(3*A001082(n)+1). - Zak Seidov, Dec 12 2007
a(n) = A077043(n) - A077043(n-1). - Reinhard Zumkeller, Dec 28 2007
a(n) = A001477(n-1) + A008619(n-1). - Yosu Yurramendi, Aug 10 2008
Euler transform of length 3 sequence [2, 1, -1]. - Michael Somos, Sep 06 2008
A011655(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009
a(n) = n - 1 + ceiling(n/2). - Michael Somos, Jan 15 2011
a(n) = 3*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i), for n>1. - Bruno Berselli, Nov 17 2010
a(n) = 3*floor(n/2) + (-1)^(n+1). - Gary Detlefs, Dec 29 2011
A215879(a(n)) > 0. - Reinhard Zumkeller, Dec 28 2012 [More precisely, A215879 is the characteristic function of A001651. - M. F. Hasler, Apr 07 2015]
a(n) = 2n - 1 - floor(n/2). - Wesley Ivan Hurt, Oct 25 2013
a(n) = (3n - 2 + (n mod 2)) / 2. - Wesley Ivan Hurt, Mar 31 2014
a(n) = A000217(n) - A000982(n-1). - Bui Quang Tuan, Mar 28 2015
1/1^3 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + ... = 4 Pi^3/(3 sqrt(3)). - M. F. Hasler, Mar 29 2015
E.g.f.: (4 + sinh(x) - cosh(x) + 3*(2*x - 1)*exp(x))/4. - Ilya Gutkovskiy, May 24 2016
a(n) = a(n+k-1) + a(n-k) - a(n-1) for n > k >= 0. - Bob Selcoe, Feb 03 2017
a(n) = -a(1-n) for all n in Z. - Michael Somos, Jul 31 2018
a(n) = n + A004526(n-1). - David James Sycamore, Sep 06 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Dec 04 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1.
Product_{n>=2} (1 + (-1)^n/a(n)) = 2*Pi/(3*sqrt(3)) (A248897). (End)

This is a list, so the offset should be 1. I corrected this and adjusted some of the comments and formulas. Other lines probably also need to be adjusted. - N. J. A. Sloane, Jan 01 2011

Offset of pre-2011 formulas verified or corrected by M. F. Hasler, Apr 07-18 2015 and by Danny Rorabaugh, Oct 23 2015

A003114 Number of partitions of n into parts 5k+1 or 5k+4.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41, 46, 54, 61, 70, 79, 91, 102, 117, 131, 149, 167, 189, 211, 239, 266, 299, 333, 374, 415, 465, 515, 575, 637, 709, 783, 871, 961, 1065, 1174, 1299, 1429, 1579, 1735, 1913, 2100, 2311, 2533, 2785

Offset: 0

N. J. A. Sloane and Herman P. Robinson

Expansion of Rogers-Ramanujan function G(x) in powers of x.
Same as number of partitions into distinct parts where the difference between successive parts is >= 2.
As a formal power series, the limit of polynomials S(n,x): S(n,x)=sum(T(i,x),0<=i<=n); T(i,x)=S(i-2,x).x^i; T(0,x)=1,T(1,x)=x; S(n,1)=A000045(n+1), the Fibonacci sequence. - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001
The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n^2)/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-1))*(1-t^(5*n-4))).
Coefficients in expansion of permanent of infinite tridiagonal matrix:
  1 1 0 0 0 0 0 0 ...
  x 1 1 0 0 0 0 0 ...
  0 x^2 1 1 0 0 0 ...
  0 0 x^3 1 1 0 0 ...
  0 0 0 x^4 1 1 0 ...
  ................... - Vladeta Jovovic, Jul 17 2004
Also number of partitions of n such that the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Jul 17 2004
Also number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Feb 27 2006
Also number of partitions of n such that if k is the largest part, then k occurs at least k times. Example: a(9)=5 because we have [3, 3, 3], [2, 2, 2, 2, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 16 2006
a(n) = number of NW partitions of n, for n >= 1; see A237981.
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[1](x). - N. J. A. Sloane, Nov 22 2015
Convolution of A109700 and A109697. - Vaclav Kotesovec, Jan 21 2017

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
G.f. = 1/q + q^59 + q^119 + q^179 + 2*q^239 + 2*q^299 + 3*q^359 + 3*q^419 + ...
From _Joerg Arndt_, Dec 27 2012: (Start)
The a(16)=17 partitions of 16 where all parts are 1 or 4 (mod 5) are
  [ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
  [ 2]  [ 4 1 1 1 1 1 1 1 1 1 1 1 1 ]
  [ 3]  [ 4 4 1 1 1 1 1 1 1 1 ]
  [ 4]  [ 4 4 4 1 1 1 1 ]
  [ 5]  [ 4 4 4 4 ]
  [ 6]  [ 6 1 1 1 1 1 1 1 1 1 1 ]
  [ 7]  [ 6 4 1 1 1 1 1 1 ]
  [ 8]  [ 6 4 4 1 1 ]
  [ 9]  [ 6 6 1 1 1 1 ]
  [10]  [ 6 6 4 ]
  [11]  [ 9 1 1 1 1 1 1 1 ]
  [12]  [ 9 4 1 1 1 ]
  [13]  [ 9 6 1 ]
  [14]  [ 11 1 1 1 1 1 ]
  [15]  [ 11 4 1 ]
  [16]  [ 14 1 1 ]
  [17]  [ 16 ]
The a(16)=17 partitions of 16 where successive parts differ by at least 2 are
  [ 1]  [ 7 5 3 1 ]
  [ 2]  [ 8 5 3 ]
  [ 3]  [ 8 6 2 ]
  [ 4]  [ 9 5 2 ]
  [ 5]  [ 9 6 1 ]
  [ 6]  [ 9 7 ]
  [ 7]  [ 10 4 2 ]
  [ 8]  [ 10 5 1 ]
  [ 9]  [ 10 6 ]
  [10]  [ 11 4 1 ]
  [11]  [ 11 5 ]
  [12]  [ 12 3 1 ]
  [13]  [ 12 4 ]
  [14]  [ 13 3 ]
  [15]  [ 14 2 ]
  [16]  [ 15 1 ]
  [17]  [ 16 ]
(End)

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109, 238.
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(e), p. 591.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 107.
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 90-92.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 290-291.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 14.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums, arXiv:hep-th/0505097, 2005.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, The Ramanujan Journal 29.1-3 (2012): 199-211.
I. Martinjak, D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities.
Mingjia Yang, Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.

Cf. A003106, A003116, A127836, A003113, A006141, A039899, A039900.
Cf. A188216 (least part k occurs at least k times).
Cf. A047209, A203776, A237981.
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
Row sums of A268187.

a003114 = p a047209_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jan 05 2011

a003114 = p 1 where
   p _ 0 = 1
   p k m = if k > m then 0 else p (k + 2) (m - k) + p (k + 1) m
-- Reinhard Zumkeller, Feb 19 2013

g:=sum(x^(k^2)/product(1-x^j,j=1..k),k=0..10): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=0..60); # Emeric Deutsch, Feb 27 2006

CoefficientList[ Series[Sum[x^k^2/Product[1 - x^j, {j, 1, k}], {k, 0, 10}], {x, 0, 65}], x][[1 ;; 61]] (* Jean-François Alcover, Apr 08 2011, after Emeric Deutsch *)
Table[Count[IntegerPartitions[n], p_ /; Min[p] >= Length[p]], {n, 0, 24}] (* Clark Kimberling, Feb 13 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 0, 0, -1, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 17 2015 *)
nmax = 60; kmax = nmax/5;
s = Flatten[{Range[0, kmax]*5 + 1}~Join~{Range[0, kmax]*5 + 4}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

{a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1), n))}; /* Michael Somos, Oct 15 2008 */

G.f.: Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^i).
The g.f. above is the special case D=2 of sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ), the g.f. for partitions into distinct part where the difference between successive parts is >= D. - Joerg Arndt, Mar 31 2014
G.f.: 1 + sum(i=1, oo, x^(5i+1)/prod(j=1 or 4 mod 5 and j<=5i+1, 1-x^j) + x^(5i+4)/prod(j=1 or 4 mod 5 and j<=5i+4, 1-x^j)). - Jon Perry, Jul 06 2004
G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2) / (Product_{i=1..k} 1 - x^(4*i))). - Michael Somos, Oct 19 2006
Euler transform of period 5 sequence [ 1, 0, 0, 1, 0, ...]. - Michael Somos, Oct 15 2008
Expansion of f(-x^5) / f(-x^1, -x^4) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, May 17 2015
Expansion of f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 13 2015
a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * 5^(1/2) * n^(3/4)) * (1 - (3*sqrt(15)/(16*Pi) + Pi/(60*sqrt(15))) / sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 23 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284150(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

A003106 Number of partitions of n into parts 5k+2 or 5k+3.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 16, 20, 22, 26, 29, 35, 38, 45, 50, 58, 64, 75, 82, 95, 105, 120, 133, 152, 167, 190, 210, 237, 261, 295, 324, 364, 401, 448, 493, 551, 604, 673, 739, 820, 899, 997, 1091, 1207, 1321, 1457, 1593, 1756, 1916, 2108, 2301

Offset: 0

N. J. A. Sloane, Herman P. Robinson

Expansion of Rogers-Ramanujan function H(x) in powers of x.
Also number of partitions of n such that the number of parts is greater by one than the smallest part. - Vladeta Jovovic, Mar 04 2006
Example: a(10)=4 because we have [9, 1], [6, 2, 2], [5, 3, 2] and [4, 4, 2]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that if the largest part is k, then there are exactly k-1 parts equal to k. Example: a(10)=4 because we have [3, 3, 2, 2], [3, 3, 2, 1, 1], [3, 3, 1, 1, 1, 1] and [2, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that if the largest part is k, then k occurs at least k+1 times. Example: a(10)=4 because we have [2, 2, 2, 2, 2], [2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n such that the smallest part is larger than the number of parts. Example: a(10)=4 because we have [10], [7, 3], [6, 4] and [5, 5]. - Emeric Deutsch, Apr 09 2006
Also number of partitions into distinct parts where parts differ by at least 2 and with minimal part >= 2, a(0)=1 because the condition is void for the empty list. - Joerg Arndt, Jan 04 2011
The g.f. is the special case D=2 of Sum_{n>=0} x^(D*n*(n+1)/2) / Product_{k=1..n} (1-x^k), the g.f. or partitions into distinct parts where the difference between successive parts is >= D and the minimal part >= D. - Joerg Arndt, Mar 31 2014
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[2](x). - N. J. A. Sloane, Nov 22 2015
Convolution of A109699 and A109698. - Vaclav Kotesovec, Jan 21 2017

			G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + 4*x^11 + ...
G.f. = q^11 + q^131 + q^191 + q^251 + q^311 + 2*q^371 + 2*q^431 + 3*q^491 + 3*q^551 + ...
From _Joerg Arndt_, Dec 27 2012: (Start)
The a(18)=15: the partitions of 18 where all parts are 2 or 3 (mod 5) are
[ 1]  [ 2 2 2 2 2 2 2 2 2 ]
[ 2]  [ 3 3 2 2 2 2 2 2 ]
[ 3]  [ 3 3 3 3 2 2 2 ]
[ 4]  [ 3 3 3 3 3 3 ]
[ 5]  [ 7 3 2 2 2 2 ]
[ 6]  [ 7 3 3 3 2 ]
[ 7]  [ 7 7 2 2 ]
[ 8]  [ 8 2 2 2 2 2 ]
[ 9]  [ 8 3 3 2 2 ]
[10]  [ 8 7 3 ]
[11]  [ 8 8 2 ]
[12]  [ 12 2 2 2 ]
[13]  [ 12 3 3 ]
[14]  [ 13 3 2 ]
[15]  [ 18 ]
(End)
From _Wolfdieter Lang_, Oct 29 2016: (Start)
The a(18)=15 partitions of 18 without part 1 and parts differing by at least 2 are:
  [18]; [16,2], [15,3], [14,4], [13,5], [12,6], [11,7], [10,8]; [12,4,2], [11,5,2], [10,6,2], [9,7,2],[10,5,3], [9,6,3], [8,6,4]. The semicolon separates different number of parts. The maximal number of parts is A259361(18) = 3. (End)

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 238.
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(f), p. 591.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 108.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 290-291.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums, arXiv:hep-th/0505097, 2005.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, arXiv:1205.6570 [math.CO], 2012; The Ramanujan Journal 29.1-3 (2012): 199-211.
I. Martinjak, D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Cf. A003114.
Cf. A047221, A219607.
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.

a003106 = p a047221_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 30 2012

g:=1/product((1-x^(5*j-2))*(1-x^(5*j-3)),j=1..15): gser:=series(g,x=0,66): seq(coeff(gser,x,n),n=0..63); # Emeric Deutsch, Apr 09 2006

max = 63; g[x_] := 1/Product[(1-x^(5j-2))*(1-x^(5j-3)), {j, 1, Floor[max/4]}]; CoefficientList[ Series[g[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 17 2011, after Emeric Deutsch *)
Table[Count[IntegerPartitions[n], p_ /; Min[p] > Length[p]], {n, 40}] (* Clark Kimberling, Feb 13 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{0, -1, -1, 0, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 17 2015 *)
nmax = 63; kmax = nmax/5;
s = Flatten[{Range[0, kmax]*5 + 2}~Join~{Range[0, kmax]*5 + 3}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

{a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, (sqrtint(4*n + 1) - 1) \ 2, t *= x^(2*k) / (1 - x^k) * (1 + x * O(x^(n - k^2 - k))), 1), n))}; /* Michael Somos, Oct 15 2008 */

The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n*(n+1))/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-2))*(1-t^(5*n-3))); this is the g.f. for the sequence.
G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2 + 2*k) / (Product_{i=1..k} 1 - x^(4*i))). - Michael Somos, Oct 19 2006
Euler transform of period 5 sequence [ 0, 1, 1, 0, 0, ...]. - Michael Somos, Oct 15 2008
From Joerg Arndt, Oct 10 2012: (Start)
Bill Gosper gives (message to the math-fun mailing list, Oct 07 2012)
prod(k>=0, [0 , a;  q^k, 1]) = [0, X(a,q);  0, Y(a,q)] where
X(a,q) = a * sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^n) ) and
Y(a,q) = sum(n>=0, a^n*q^(n^2-n) / prod(k=1..n, 1-q^n) ).
Set a=q to obtain prod(k>=0, [0 , a;  q^k, 1]) = [0, q*H(q);  0, G(q)] where
H(q) is the g.f. of A003106 and G(q) is the g.f. of A003114.
Bill Gosper and N. J. A. Sloane give (message to math-fun, Oct 10 2012)
prod(k>=0, [0 , a*q^k;  1, 1]) = [U(a,q), U(a,q);  V(a,q), V(a,q)] where
U(a,q) = a * sum(n>=0, a^n*q^(n^2+n) / prod(k=1..n, 1-q^k) ) and
V(a,q) = sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^k) ).
Set a=1 to obtain prod(k>=0, [0 , q^k;  1, 1]) = [H(q), H(q);  G(q), G(q)].
(End)
Expansion of f(-x^5) / f(-x^2, -x^3) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, May 06 2015
Expansion of f(-x, -x^4) / f(-x) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, Jun 13 2015
a(n) ~ sqrt((sqrt(5)-1)/5) * exp(2*Pi*sqrt(n/15)) / (2^(3/2) * 3^(1/4) * n^(3/4)) * (1 + (11*Pi/(60*sqrt(15)) - 3*sqrt(15)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 24 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284152(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

A000726 Number of partitions of n in which no parts are multiples of 3.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 57, 70, 89, 108, 135, 163, 202, 243, 297, 355, 431, 513, 617, 731, 874, 1031, 1225, 1439, 1701, 1991, 2341, 2731, 3197, 3717, 4333, 5022, 5834, 6741, 7803, 8991, 10375, 11923, 13716, 15723, 18038, 20628, 23603

Offset: 0

N. J. A. Sloane

Case k=4, i=3 of Gordon Theorem.
Expansion of q^(-1/12)*eta(q^3)/eta(q) in powers of q. - Michael Somos, Apr 20 2004
Euler transform of period 3 sequence [1,1,0,...]. - Michael Somos, Apr 20 2004
Also the number of partitions with at most 2 parts of size 1 and all differences between parts at distance 3 are greater than 1. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,2] (for example, [2,2,1,1] does not qualify because the difference between the first and the fourth parts is equal to 1). - Emeric Deutsch, Apr 18 2006
Also the number of partitions of n where no part appears more than twice. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,1,1]. - Emeric Deutsch, Apr 18 2006
Also the number of partitions of n with least part either 1 or 2 and with differences of consecutive parts at most 2. Example: a(6)=7 because we have [4,2], [3,2,1], [3,1,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1] and [1,1,1,1,1,1]. - Emeric Deutsch, Apr 18 2006
Equals left border of triangle A174714. - Gary W. Adamson, Mar 27 2010
Triangle A113685 is equivalent to p(x) = p(x^2) * A000009(x); given A000041(x) = p(x). Triangle A176202 is equivalent to p(x) = p(x^3) * A000726(x). - Gary W. Adamson, Apr 11 2010
Convolution of A035382 and A035386. - Vaclav Kotesovec, Aug 23 2015
The number of partitions of n in which no parts are multiples of k equals the number of partitions of n where no part appears more than k-1 times. - Gregory L. Simay, Oct 15 2022

			There are a(6)=7 partitions of 6 into parts != 0 (mod 3):
[ 1]  [5,1],
[ 2]  [4,2],
[ 3]  [4,1,1],
[ 4]  [2,2,2],
[ 5]  [2,2,1,1],
[ 6]  [2,1,1,1,1], and
[ 7]  [1,1,1,1,1,1]
.
From _Joerg Arndt_, Dec 29 2012: (Start)
There are a(10)=22 partitions p(1)+p(2)+...+p(m)=10 such that p(k)!=p(k-2) (that is, no part appears more than twice):
[ 1]  [ 3 3 2 1 1 ]
[ 2]  [ 3 3 2 2 ]
[ 3]  [ 4 2 2 1 1 ]
[ 4]  [ 4 3 2 1 ]
[ 5]  [ 4 3 3 ]
[ 6]  [ 4 4 1 1 ]
[ 7]  [ 4 4 2 ]
[ 8]  [ 5 2 2 1 ]
[ 9]  [ 5 3 1 1 ]
[10]  [ 5 3 2 ]
[11]  [ 5 4 1 ]
[12]  [ 5 5 ]
[13]  [ 6 2 1 1 ]
[14]  [ 6 2 2 ]
[15]  [ 6 3 1 ]
[16]  [ 6 4 ]
[17]  [ 7 2 1 ]
[18]  [ 7 3 ]
[19]  [ 8 1 1 ]
[20]  [ 8 2 ]
[21]  [ 9 1 ]
[22]  [ 10 ]
(End)

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145.
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 and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See p. 79.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
Eric Weisstein's World of Mathematics, Partition function b_k.
Wikipedia, Glaisher's Theorem.

Cf. A000009 (no multiples of 2), A001935 (no of 4), A035959 (no of 5), A219601 (no of 6), A035985, A001651, A003105, A035361, A035360.
Cf. A174714. - Gary W. Adamson, Mar 27 2010
Cf. A113685, A176202. - Gary W. Adamson, Apr 11 2010
Cf. A046913.
Column k=3 of A286653.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a000726 n = p a001651_list n where
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 23 2011

g:=product(1+x^j+x^(2*j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..50); # Emeric Deutsch, Apr 18 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(irem(d, 3)=0, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 17 2017

f[0] = 1; f[n_] := Coefficient[Expand@ Product[1 + x^k + x^(2k), {k, n}], x^n]; Table[f@n, {n, 0, 40}] (* Robert G. Wilson v, Nov 10 2006 *)
QP = QPochhammer; CoefficientList[QP[q^3]/QP[q] + O[q]^60, q] (* Jean-François Alcover, Nov 24 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 02 2016 *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 3], 0, 2] ], {n, 0, 50}] (* Robert Price, Jul 28 2020 *)
Table[Count[IntegerPartitions[n],?(NoneTrue[Mod[#,3]==0&])],{n,0,50}] (* _Harvey P. Dale, Sep 06 2022 *)

a(n)=if(n<0,0,polcoeff(eta(x^3+x*O(x^n))/eta(x+x*O(x^n)),n))

lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q))} \\ Altug Alkan, Mar 20 2018

G.f.: 1/(Product_{k>=1} (1-x^(3*k-1))*(1-x^(3*k-2))) = Product_{k>=1} (1 + x^k + x^(2*k)) (where 1 + x + x^2 is the 3rd cyclotomic polynomial).
a(n) = A061197(n, n).
Given g.f. A(x) then B(x) = x*A(x^6)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u,v,w) = +v^2 +v*w^2 -v*u^2 +3*u^2*w^2. - Michael Somos, May 28 2006
G.f.: P(x^3)/P(x) where P(x) = Product_{k>=1} (1 - x^k). - Joerg Arndt, Jun 21 2011
a(n) ~ 2*Pi * BesselI(1, sqrt((12*n + 1)/3)*Pi/3) / (3*sqrt(12*n + 1)) ~ exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)) * (1 + (Pi/36 - 9/(16*Pi))/sqrt(n) + (Pi^2/2592 - 135/(512*Pi^2) - 5/64)/n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 13 2017
a(n) = (1/n)*Sum_{k=1..n} A046913(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^(3*k)))). - Ilya Gutkovskiy, Aug 15 2018

More terms from Olivier Gérard

A003108 Number of partitions of n into cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 21, 21, 21, 22, 22, 22, 23, 23, 25, 26, 26, 27, 27, 27, 28

Offset: 0

N. J. A. Sloane, Herman P. Robinson

nonn

The g.f. 1/(z+1)/(z**2+1)/(z**4+1)/(z-1)**2 conjectured by Simon Plouffe in his 1992 dissertation is wrong.

			a(16) = 3 because we have [8,8], [8,1,1,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 +...
such that the g.f. A(x) satisfies the identity [Paul D. Hanna]:
A(x) = 1/((1-x)*(1-x^8)*(1-x^27)*(1-x^64)*(1-x^125)*...)
A(x) = 1 + x/(1-x) + x^8/((1-x)*(1-x^8)) + x^27/((1-x)*(1-x^8)*(1-x^27)) + x^64/((1-x)*(1-x^8)*(1-x^27)*(1-x^64)) +...

H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..100000 (terms 0..1000 from T. D. Noe)
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.
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
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Partition
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A000578, A068980, A131799, A218495, A226748, A279329, A280263.
Cf. A001156, A046042.
Cf. A037444, A259792, A259793.

a003108 = p $ tail a000578_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Oct 31 2012

[#RestrictedPartitions(n,{d^3:d in [1..n]}): n in [0..150]]; // Marius A. Burtea, Jan 02 2019

g:=1/product(1-x^(j^3),j=1..30): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..65); # Emeric Deutsch, Mar 30 2006

nmax = 100; CoefficientList[Series[Product[1/(1 - x^(k^3)), {k, 1, nmax^(1/3)}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
nmax = 60; cmax = nmax^(1/3);
s = Table[n^3, {n, cmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

{a(n)=polcoeff(1/prod(k=1, ceil(n^(1/3)), 1-x^(k^3)+x*O(x^n)), n)} /* Paul D. Hanna, Mar 09 2012 */

{a(n)=polcoeff(1+sum(m=1, ceil(n^(1/3)), x^(m^3)/prod(k=1, m, 1-x^(k^3)+x*O(x^n))), n)} /* Paul D. Hanna, Mar 09 2012 */

from functools import lru_cache
from sympy import integer_nthroot, divisors
@lru_cache(maxsize=None)
def A003108(n):
    @lru_cache(maxsize=None)
    def a(n): return integer_nthroot(n,3)[1]
    @lru_cache(maxsize=None)
    def c(n): return sum(d for d in divisors(n,generator=True) if a(d))
    return (c(n)+sum(c(k)*A003108(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

G.f.: 1/Product_{j>=1} (1-x^(j^3)). - Emeric Deutsch, Mar 30 2006
G.f.: Sum_{n>=0} x^(n^3) / Product_{k=1..n} (1 - x^(k^3)). - Paul D. Hanna, Mar 09 2012
a(n) ~ exp(4 * (Gamma(1/3)*Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3)*Zeta(4/3))^(3/4) / (24*Pi^2*n^(5/4)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

A003107 Number of partitions of n into Fibonacci parts (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 14, 17, 22, 27, 33, 41, 49, 59, 71, 83, 99, 115, 134, 157, 180, 208, 239, 272, 312, 353, 400, 453, 509, 573, 642, 717, 803, 892, 993, 1102, 1219, 1350, 1489, 1640, 1808, 1983, 2178, 2386, 2609, 2854, 3113, 3393, 3697, 4017, 4367, 4737

Offset: 0

N. J. A. Sloane, Herman P. Robinson

The partitions allow repeated items but the order of items is immaterial (1+2=2+1). - Ron Knott, Oct 22 2003

A098641(n) = a(A000045(n)). - Reinhard Zumkeller, Apr 24 2005

			a(4) = 4 since the 4 partitions of 4 using only Fibonacci numbers, repetitions allowed, are 1+1+1+1, 2+2, 2+1+1, 3+1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
G. Almkvist, Partitions with Parts in a Finite Set and with Parts Outside a Finite Set, Exper. Math. vol 11 no 4 (2002) p 449-456.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.

Cf. A007000, A005092, A028290 (where the only Fibonacci numbers allowed are 1, 2, 3, 5 and 8).
Cf. A000045, A000119, A102848, A238998.
Row sums of A319394.

import Data.MemoCombinators (memo2, integral)
a003107 n = a003107_list !! n
a003107_list = map (p' 2) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m | m < fib   = 0
         | otherwise = p' k (m - fib) + p' (k + 1) m where fib = a000045 k
-- Reinhard Zumkeller, Dec 09 2015

F:= combinat[fibonacci]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
       b(n, i-1)+`if`(F(i)>n, 0, b(n-F(i), i))))
    end:
a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)
       while F(j+1)<=n do od; b(n, j)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2013

CoefficientList[ Series[1/ Product[1 - x^Fibonacci[i], {i, 2, 21}], {x, 0, 53}], x] (* Robert G. Wilson v, Mar 28 2006 *)
nmax = 53;
s = Table[Fibonacci[n], {n, nmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)
F = Fibonacci;
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 2, 0,
     b[n, i - 1] + If[F[i] > n, 0, b[n - F[i], i]]]];
a[n_] := Module[{j}, For[j = Floor@Log[(1+Sqrt[5])/2, n+1],
     F[j + 1] <= n, j++]; b[n, j]];
a /@ Range[0, 100] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

f(x,y,z)=if(xCharles R Greathouse IV, Dec 14 2015

a(n) = (1/n)*Sum_{k=1..n} A005092(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Jan 21 2002
G.f.: Product_{i>=2} 1/(1-x^fibonacci(i)). - Ron Knott, Oct 22 2003
a(n) = f(n,1,1) with f(x,y,z) = if xReinhard Zumkeller, Nov 11 2009
G.f.: 1 + Sum_{i>=2} x^Fibonacci(i) / Product_{j=2..i} (1 - x^Fibonacci(j)). - Ilya Gutkovskiy, May 07 2017

More terms from Vladeta Jovovic, Jan 21 2002

A098151 Number of partitions of 2*n with no part divisible by 3 and all odd parts occurring with even multiplicities.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 24, 36, 52, 74, 104, 144, 198, 268, 360, 480, 634, 832, 1084, 1404, 1808, 2316, 2952, 3744, 4728, 5946, 7448, 9294, 11556, 14320, 17688, 21780, 26740, 32736, 39968, 48672, 59122, 71644, 86616, 104484, 125768, 151072, 181104, 216684

Offset: 0

Noureddine Chair, Aug 29 2004

There are no partitions of 2n+1 in which all odd parts occur with even multiplicity. - Michael Somos, Apr 15 2012
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is also the number of Schur overpartitions of n, i.e., the number of overpartitions of n where adjacent parts differ by at least 3 if the smaller is overlined or divisible by 3 and adjacent parts differ by at least 6 if the smaller is overlined and divisible by 3. - Jeremy Lovejoy, Mar 23 2015
Let A(q) denote the g.f. of this sequence. Let m be a nonzero integer. The simple continued fraction expansions of the real numbers A(1/(2*m)) and A(1/(2*m+1)) may be predictable. For a given positive integer n, the sequence of the n-th partial denominators of the continued fractions are conjecturally polynomial or quasi-polynomial in m for m sufficiently large. An example is given below. Cf. A080054. - Peter Bala, Jun 09 2025

			a(4)=10 because 8 = 4+4 = 4+2+2=2+2+2+2 = 2+2+2+1+1 = 2+2+1+1+1+1 = 4+2+1+1 = 4+1+1+1+1 = 2+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1.
G.f. = 1 + 2*q + 4*q^2 + 6*q^3 + 10*q^4 + 16*q^5 + 24*q^6 + 36*q^7 + 52*q^8 + ...
From _Peter Bala_, Jun 09 2025: (Start)
G.f.: A(q) = f(q, q^2) / f(-q, -q^2).
Simple continued fraction expansions of A(1/(2*m)):
m =  2  [1;  1   9  1    5    8    45   4  1  2  1  1  1  3  3   2  2 ...]
m =  3  [1;  2  13  1   14   12   133   8  1  1  7  2  1  2  2   1  1 ...]
m =  4  [1;  3  17  1   27   16   297  12  2  2  1  1  1  2  2   2  2 ...]
m =  5  [1;  4  21  1   44   20   561  16  2  1  7  3  3  2  2  25  8 ...]
m =  6  [1;  5  25  1   65   24   949  20  3  2  1  1  1  3  4   2  1 ...]
m =  7  [1;  6  29  1   90   28  1485  24  3  1  7  4  5  2  1   1  6 ...]
m =  8  [1;  7  33  1  119   32  2193  28  4  2  1  1  1  4  6   2  1 ...]
m =  9  [1;  8  37  1  152   36  3097  32  4  1  7  5  7  2  1   1  3 ...]
m = 10  [1;  9  41  1  189   40  4221  36  5  2  1  1  1  5  8   2  1 ...]
...
The sequence of the 4th partial denominators [5, 14, 27, 44, ...] appears to be given by the polynomial (2*m + 1)*(m - 1) for m >= 2.
The sequence of the 6th partial denominators [45, 133, 297, 561, ...] appears to be given by the polynomial (2*m + 1)*(2*m^2 + 1) for m >= 2. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 14-16.
Byungchan Kim and Eunmi Kim, Partitions weighted by the number of two types of parts, Bull. Korean Math. Soc. (2024) Vol. 61, No. 6, 1677-1684. See p. 1679.
Jeremy Lovejoy, A theorem on seven-colored overpartitions and its applications, Int. J. Number Theory. 1 (2005) 215-224
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S24.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A015128, A001318, A080054, A080995, A010815, A103260, A215594.

series(product((1+x^k+x^(2*k))/(1-x^k+x^(2*k)),k=1..150),x=0,100);
# alternative program using expansion of f(q, q^2) / f(-q, -q^2):
with(gfun): series( add(x^(n*(3*n-1)/2),n = -8..8)/add((-1)^n*x^(n*(3*n-1)/2), n = -8..8), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^3]^2 / (QPochhammer[ q]^2 QPochhammer[ q^6]), {q, 0, n}] (* Michael Somos, Oct 23 2013 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1)) * (1+x^(3*k-2)) / ((1-x^(3*k-1)) * (1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))} /* Michael Somos, Dec 04 2004 */

Expansion of phi(-q^3) / phi(-q) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Apr 15 2012
Expansion of f(q, q^2) / f(-q, -q^2) in powers of q where f(,) is the Ramanujan two-variable theta function. - Michael Somos, Apr 15 2012
Expansion of eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
G.f. = (Sum_{n = -oo..oo} (-1)^n*q^(3*n^2)) / (Sum_{n = -oo..oo} (-1)^n*q^(n^2)). - N. J. A. Sloane, Aug 09 2016
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u^2) * (u^2 + v^4) - 4 * u^2*v^4. - Michael Somos, Apr 15 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3 * u*v^2 - 3 * u^2*v^3. - Michael Somos, Dec 04 2004
Euler transform of period 6 sequence [2, 1, 0, 1, 2, 0, ...]. - Vladeta Jovovic, Sep 24 2004
Taylor series of product_{k=1..inf}(1+x^k+x^(2*k))/(1-x^k+x^(2*k))= product_{k=1..inf}(1+x^k)(1-x^(3k))/((1-x^k)(1+x^(3k)))=Theta_4(0, x^3)/theta_4(0, x)
a(n) ~ Pi * BesselI(1, Pi*sqrt(2*n/3)) / (3*sqrt(2*n)) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)) * (1 - 3*sqrt(3)/(8*Pi*sqrt(2*n)) - 45/(256*Pi^2*n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 09 2017
Convolution of A000726 and A003105. - R. J. Mathar, Nov 17 2017
From Peter Bala, Jun 09 2025: (Start)
G.f.: A(q) = Sum_{n = -oo..oo} q^(n*(3*n+1)/2) / Sum_{n = -oo..oo} (-1)^n * q^(n*(3*n+1)/2).
Recurrences:
a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + + - - ... = f(n), where [0, 1, 2, 5, 7, 12, 15, ...] is the sequence of generalized pentagonal numbers A001318, a(n) is set equal to 0 for negative n and f(n) = 1 if n is a generalized pentagonal number, otherwise f(n) = 0 (see A080995). Compare with the recurrence for the partition function p(n) = A000041(n).
a(n) - 2*a(n-1) + 2*a(n-4) - 2*a(n-9) + 2*a(n-16) - 2*a(n-25) + - ... = g(n), where g(n) = 2*(-1)^k if n is of the form 3*(k^2), otherwise g(n) = 0. (End)

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A285219 Indices of primes in A003105.

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A304047 Numbers k such that A003105(k) is divisible by k.

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A002865 Number of partitions of n that do not contain 1 as a part.

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A001651 Numbers not divisible by 3.

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A003114 Number of partitions of n into parts 5k+1 or 5k+4.

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A003106 Number of partitions of n into parts 5k+2 or 5k+3.

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