A103221 -id:A103221 - cp's OEIS frontend

A016921 a(n) = 6*n + 1.

1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325, 331

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 22 ).
Also solutions to 2^x + 3^x == 5 (mod 7). - Cino Hilliard, May 10 2003
Except for 1, exponents n > 1 such that x^n - x^2 - 1 is reducible. - N. J. A. Sloane, Jul 19 2005
Let M(n) be the n X n matrix m(i,j) = min(i,j); then the trace of M(n)^(-2) is a(n-1) = 6*n - 5. - Benoit Cloitre, Feb 09 2006
If Y is a 3-subset of an (2n+1)-set X then, for n >= 3, a(n-1) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
All composite terms belong to A269345 as shown in there. - Waldemar Puszkarz, Apr 13 2016
First differences of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
For b(n) = A103221(n) one has b(a(n)-1) = b(a(n)+1) = b(a(n)+2) = b(a(n)+3) = b(a(n)+4) = n+1 but b(a(n)) = n. So-called "dips" in A103221. See the Avner and Gross remark on p. 178. - Wolfdieter Lang, Sep 16 2016
A (n+1,n) pebbling move involves removing n + 1 pebbles from a vertex in a simple graph and placing n pebbles on an adjacent vertex. A two-player impartial (n+1,n) pebbling game involves two players alternating (n+1,n) pebbling moves. The first player unable to make a move loses. The sequence a(n) is also the minimum number of pebbles such that any assignment of those pebbles on a complete graph with 3 vertices is a next-player winning game in the two player impartial (k+1,k) pebbling game. These games are represented by A347637(3,n). - Joe Miller, Oct 18 2021
Interleaving of A017533 and A017605. - Leo Tavares, Nov 16 2021

			From _Ilya Gutkovskiy_, Apr 15 2016: (Start)
Illustration of initial terms:
                      o
                    o o o
              o     o o o
            o o o   o o o
      o     o o o   o o o
    o o o   o o o   o o o
o   o o o   o o o   o o o
n=0  n=1     n=2     n=3
(End)

Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kayla Barker, Mia DeStefano, Eugene Fiorini, Michael Gohn, Joe Miller, Jacob Roeder, and Tony W. H. Wong, Generalized Impartial Two-player Pebbling Games on K_3 and C_4, J. Int. Seq. (2024) Vol. 27, Issue 5, Art. No. 24.5.8. See p. 3.
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Leo Tavares, Illustration: Hexagonal Lines.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093563 ((6, 1) Pascal, column m=1).
Cf. A000567 (partial sums), A002476 (primes), A005408, A008588, A016813, A016933, A016945, A016957, A017281, A017533, A128470, A158057, A161700, A161705, A161709, A161714.
a(n) = A007310(2*(n+1)); complement of A016969 with respect to A007310.
Cf. A287326 (second column).
Cf. A000217, A003215.
Cf. A001477, A017605.

List([0..60], n-> 6*n+1); # G. C. Greubel, Sep 18 2019

a016921 = (+ 1) . (* 6)
a016921_list = [1, 7 ..]  -- Reinhard Zumkeller, Jan 15 2013

[6*n+1: n in [0..60]]; // G. C. Greubel, Sep 18 2019

a[1]:=1:for n from 2 to 100 do a[n]:=a[n-1]+6 od: seq(a[n], n=1..56); # Zerinvary Lajos, Mar 16 2008

Range[1, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

a(n)=6*n+1 \\ Charles R Greathouse IV, Mar 22 2016

for n in range(0,10**5):print(6*n+1) # Soumil Mandal, Apr 14 2016

a(n) = 6*n + 1, n >= 0 (see the name).
G.f.: (1+5*x)/(1-x)^2.
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A013730(n). - Reinhard Zumkeller, Feb 24 2009
a(n) = 4*(3*n-1) - a(n-1) (with a(0)=1). - Vincenzo Librandi, Nov 20 2010
E.g.f.: (1 + 6*x)*exp(x). - G. C. Greubel, Sep 18 2019
a(n) = A003215(n) - 6*A000217(n-1). See Hexagonal Lines illustration. - Leo Tavares, Sep 10 2021
From Leo Tavares, Oct 27 2021: (Start)
a(n) = 6*A001477(n-1) + 7
a(n) = A016813(n) + 2*A001477(n)
a(n) = A017605(n-1) + A008588(n-1)
a(n) = A016933(n) - 1
a(n) = A008588(n) + 1. (End)
Sum_{n>=0} (-1)^n/a(n) = Pi/6 + sqrt(3)*arccoth(sqrt(3))/3. - Amiram Eldar, Dec 10 2021

A008615 a(n) = floor(n/2) - floor(n/3).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 13, 14, 14, 14, 14, 15, 14

Offset: 0

N. J. A. Sloane, Simon Plouffe

If the two leading 0's are dropped, this becomes the essentially identical sequence A103221, with g.f. 1/((1-x^2)*(1-x^3)), which arises in many contexts. For example, 1/((1-x^4)*(1-x^6)) is the Poincaré series [or Poincare series] for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).
Dimension of the space of weight 2n+8 cusp forms for Gamma_0( 1 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
a(n) is the number of ways n can be written as the sum of a positive even number and a nonnegative multiple of 3 and so the number of ways (n-2) can be written as the sum of a nonnegative even number and a nonnegative multiple of 3 and also the number of ways (n+3) can be written as the sum of a positive even number and a positive multiple of 3.
It appears that this is also the number of partitions of 2n+6 that are 4-term arithmetic progressions. - John W. Layman, May 01 2009 [verified by Wesley Ivan Hurt, Jan 17 2021]
a(n) is the number of (n+3)-digit fixed points under the base-3 Kaprekar map A164993 (see A164997 for the list of fixed points). - Joseph Myers, Sep 04 2009
Starting from n=10 also the number of balls in new consecutive hexagonal edges, if an (infinite) chain of balls is winded spirally around the first ball at the center, such that each six steps make an entire winding. - K. G. Stier, Dec 21 2012
In any three consecutive terms at least two of them are equal to each other. - Michael Somos, Mar 01 2014
Number of partitions of (n-2) into parts 2 and 3. - David Neil McGrath, Sep 05 2014
a(n), n >= 0, is also the dimension of S_{2*(n+4)}, the complex vector space of modular cusp forms of weight 2*(n+4) and level 1 (full modular group). The dimension of S_0, S_2, S_4 and S_6 is 0. See, e.g., Ash and Gross, p. 178. Table 13.1. - Wolfdieter Lang, Sep 16 2016
From Wolfdieter Lang, May 08 2017: (Start)
a(n-2) = floor((n-2)/2) - floor((n-2)/3) = floor(n/2) - floor((n+1)/3) is for n >=0 the number of integers k in the interval (n+1)/3 < k <= floor(n/2). This problem appears in the computation of the number of zeros of Chebyshev S(n, x) polynomials (coefficients in A049310) in the open interval (-1, +1). See a comment there. This computation was motivated by a conjecture given in A008611 by Michel Lagneau, Mar 31 2017.
a(n) is also the number of integers k in the closed interval (n+1)/3 <= k <= floor(n/2), which is floor(n/2) - (ceiling((n+1)/3) - 1) for n >= 0 (proof trivial for n+1 == 0 (mod 3) and otherwise). From the preceding statement this a(n) is also a(n-2) + [n == 2 (mod 3)] for n >= 0 (with [statement] = 1 if the statement is true and zero otherwise). This proves the recurrence given by Michael Somos in the formula section. (End)
Assuming the Collatz conjecture to be true, for n > 1, a(n+7) is the row length of the n-th row of A340985. That is, the number of weakly connected components of the Collatz digraph of order n. - Sebastian Karlsson, Feb 23 2021

			G.f. = x^2 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + ...

Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
J.-M. Kantor, Où en sont les mathématiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
David Broadhurst, Feynman integrals, L-series and Kloosterman moments, arXiv:1604.03057 [physics.gen-ph], 2016. See Cor. 1.
J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 212
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 448
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 20.
T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 1967.
William A. Stein, The modular forms database
James Tanton, Integer Triangles, Chapter 11 in "Mathematics Galore!" (MAA, 2012).
James Tanton, Young students approach integer triangles, FOCUS 22 no. 5 (2002), 4 - 6.
James Tanton et al., Young students approach integer triangles, FOCUS 22 no. 5 (2002), 4 - 6.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Essentially the same as A103221.
First differences of A069905 (and A001399).
Cf. A002264, A004009, A004526, A005044, A008588, A008611, A013973, A016921, A016933, A016957, A049310, A057078, A164993, A164997, A340985.

a008615 n = n `div` 2 - n `div` 3  -- Reinhard Zumkeller, Apr 28 2014

[Floor(n/2)-Floor(n/3): n in [0..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

a := func< n | n lt 2 select 0 else n eq 2 select 1 else Dimension( ModularForms( PSL2( Integers()), 2*n-4))>; /* Michael Somos, Dec 11 2018 */

a := n-> floor(n/2) - floor(n/3): seq(a(n), n = 0 .. 87);

a[n_]:=Floor[n/2]-Floor[n/3]; Array[a,90,0] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008; corrected by Harvey P. Dale, Nov 30 2011 *)
LinearRecurrence[{0, 1, 1, 0, -1}, {0, 0, 1, 0, 1}, 100]; (* Vincenzo Librandi, Sep 09 2015 *)

{a(n) = (n\2) - (n\3)}; /* Michael Somos, Feb 06 2003 */

def A008615(n): return n//2 - n//3 # Chai Wah Wu, Jun 07 2022

a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - Henry Bottomley, Sep 02 2000
G.f.: x^2 / ((1-x^2) * (1-x^3)).
From Reinhard Zumkeller, Feb 27 2008: (Start)
a(A016933(n)) = a(A016957(n)) = a(A016969(n)) = n+1.
a(A008588(n)) = a(A016921(n)) = a(A016945(n)) = n. (End)
a(6*k) = k, k >= 0. - Zak Seidov, Sep 09 2012
a(n) = A005044(n+1) - A005044(n-3). - Johannes W. Meijer, Oct 18 2013
a(n) = floor((n+4)/6) - floor((n+3)/6) + floor((n+2)/6). - Mircea Merca, Nov 27 2013
Euler transform of length 3 sequence [0, 1, 1]. - Michael Somos, Mar 01 2014
a(n+2) = a(n) + 1 if n == 0 (mod 3), a(n+2) = a(n) otherwise. - Michael Somos, Mar 01 2014. See the May 08 2017 comment above. - Wolfdieter Lang, May 08 2017
a(n) = -a(-1 - n) for all n in Z. - Michael Somos, Mar 01 2014.
a(n) = A004526(n) - A002264(n). - Reinhard Zumkeller, Apr 28 2014
a(n) = Sum_{i=0..n-2} (floor(i/6)-floor((i-3)/6))*(-1)^i. - Wesley Ivan Hurt, Sep 08 2015
a(n) = a(n+6) - 1 = A103221(n+4) - 1, n >= 0. - Wolfdieter Lang, Sep 16 2016
12*a(n) = 2*n +1 +3*(-1)^n -4*A057078(n). - R. J. Mathar, Jun 19 2019
a(n) = Sum_{k=1..floor((n+3)/2)} Sum_{j=k..floor((2*n+6-k)/3)} Sum_{i=j..floor((2*n+6-j-k)/2)} ([j-k = i-j = 2*n+6-2*i-j-k] - [k = j = i = 2*n+6-i-j-k]), where [ ] is the (generalized) Iverson bracket. - Wesley Ivan Hurt, Jan 17 2021
E.g.f.: (3*(2 + x)*cosh(x) - 2*exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 3*(x-1)*sinh(x))/18. - Stefano Spezia, Oct 17 2022

A266755 Expansion of 1/((1-x^2)(1-x^3)(1-x^4)).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, 24, 21, 27, 24, 30, 27, 33, 30, 37, 33, 40, 37, 44, 40, 48, 44, 52, 48, 56, 52, 61, 56, 65, 61, 70, 65, 75, 70, 80, 75, 85, 80, 91, 85, 96, 91, 102, 96, 108, 102, 114, 108, 120, 114, 127, 120, 133, 127, 140, 133, 147, 140, 154, 147, 161, 154, 169

Offset: 0

N. J. A. Sloane, Jan 10 2016

This is the same as A005044 but without the three leading zeros. There are so many situations where one wants this sequence rather than A005044 that it seems appropriate for it to have its own entry.
But see A005044 (still the main entry) for numerous applications and references.
Also, Molien series for invariants of finite Coxeter group D_3.
The Molien series for the finite Coxeter group of type D_k (k >= 3) has g.f. = 1/Product_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence.
Also, Molien series for invariants of finite Coxeter group A_3. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k not the alternating group Alt_k.
a(n) is the number of partitions of n into parts 2, 3, and 4. - Joerg Arndt, Apr 16 2017
From Gus Wiseman, May 23 2021: (Start)
Also the number of integer partitions of n into at most n/2 parts, none greater than 3. The case of any maximum is A110618. The case of any length is A001399. The Heinz numbers of these partitions are given by A344293.
For example, the a(2) = 1 through a(13) = 5 partitions are:
    2  3  22  32  33   322  332   333   3322   3332   3333    33322
          31      222  331  2222  3222  3331   32222  33222   33331
                  321       3221  3321  22222  33221  33321   322222
                            3311        32221  33311  222222  332221
                                        33211         322221  333211
                                                      332211
                                                      333111
(End)

			G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 4*x^8 + ... - _Michael Somos_, Jan 29 2022

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sara C. Billey, Matjaž Konvalinka, and Joshua P. Swanson, Asymptotic normality of the major index on standard tableaux, arXiv:1905.00975 [math.CO], 2019.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1).

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.
Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.
A variant of A005044.
Cf. A001400 (partial sums).
Cf. A308065.
Number of partitions of n whose Heinz number is in A344293.
A001399 counts partitions with all parts <= 3, ranked by A051037.
A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
A035363 counts partitions of n with n/2 parts, ranked by A340387.
A110618 counts partitions of n into at most n/2 parts, ranked by A344291.
Cf. A000041, A008642, A279622, A325691, A344294, A344297.

I:=[1,0,1,1,2,1,3,2,4]; [n le 9 select I[n] else Self(n-2)+ Self(n-3)+Self(n-4)-Self(n-5)-Self(n-6)-Self(n-7)+Self(n-9): n in [1..100]]; // Vincenzo Librandi, Jan 11 2016

CoefficientList[Series[1/((1-x^2)(1-x^3)(1-x^4)), {x, 0, 100}], x] (* JungHwan Min, Jan 10 2016 *)
LinearRecurrence[{0,1,1,1,-1,-1,-1,0,1}, {1,0,1,1,2,1,3,2,4}, 100] (* Vincenzo Librandi, Jan 11 2016 *)
Table[Length[Select[IntegerPartitions[n],Length[#]<=n/2&&Max@@#<=3&]],{n,0,30}] (* Gus Wiseman, May 23 2021 *)
a[ n_] := Round[(n + 3*(2 - Mod[n,2]))^2/48]; (* Michael Somos, Jan 29 2022 *)

Vec(1/((1-x^2)*(1-x^3)*(1-x^4)) + O(x^100)) \\ Michel Marcus, Jan 11 2016

{a(n) = round((n + 3*(2-n%2))^2/48)}; /* Michael Somos, Jan 29 2022 */

(1/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Jun 13 2019

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8. - Vincenzo Librandi, Jan 11 2016
a(n) = a(-9-n) for all n in Z. a(n) = a(n+3) for all n in 2Z. - Michael Somos, Jan 29 2022
E.g.f.: exp(-x)*(81 - 18*x + exp(2*x)*(107 + 60*x + 6*x^2) + 64*exp(x/2)*cos(sqrt(3)*x/2) + 36*exp(x)*(cos(x) - sin(x)))/288. - Stefano Spezia, Mar 05 2023
For n >= 3, if n is even, a(n) = a(n-3) + floor(n/4) + 1, otherwise a(n) = a(n-3). - Robert FERREOL, Feb 05 2024
a(n) = floor((n^2+9*n+(3*n+9)*(-1)^n+39)/48). - Hoang Xuan Thanh, Jun 03 2025

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

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 22, 23, 28, 29, 34, 36, 42, 44, 50, 53, 60, 63, 71, 74, 83, 87, 96, 101, 111, 116, 127, 133, 145, 151, 164, 171, 185, 193, 207, 216, 232, 241, 258, 268, 286, 297, 316, 328, 348, 361, 382, 396, 419, 433, 457

Offset: 0

N. J. A. Sloane

Also, Molien series for invariants of finite Coxeter group A_4. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k not the alternating group Alt_k. - N. J. A. Sloane, Jan 11 2016

Number of partitions into parts 2, 3, 4, and 5. - Joerg Arndt, Apr 29 2014

			a(4)=2 because f''''(x)/4!=2 at x=0 for f=1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)).
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + 7*x^11 + ... .

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 32).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 241
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1,-2,-1,0,1,1,1,0,-1).

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

Cf. A005044, A001401 (partial sums).

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019

seq(coeff(series(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 08 2019

SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)),{x,0,#}]&/@Range[0,100] (* or *) a[k_]=SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4) (1-x^5)),{x,0,k}] (* Peter Pein (petsie(AT)dordos.net), Sep 09 2006 *)
CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,5}],{x,0,70}],x] (* Harvey P. Dale, Feb 22 2018 *)

{a(n) = if( n<-13, -a(-14 - n), polcoeff( prod( k=2, 5, 1 / (1 - x^k), 1 + x * O(x^n)), n))} /* Michael Somos, Oct 14 2006 */

def A008667_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))).list()
A008667_list(65) # G. C. Greubel, Sep 08 2019

Euler transform of length 5 sequence [ 0, 1, 1, 1, 1]. - Michael Somos, Sep 23 2006
a(-14 - n) = -a(n). - Michael Somos, Sep 23 2006
a(n) ~ 1/720*n^3. - Ralf Stephan, Apr 29 2014
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6) - 2*a(n-7) - a(n-8) + a(n-10) + a(n-11) + a(n-12) - a(n-14). - David Neil McGrath, Sep 13 2014
From R. J. Mathar, Jun 23 2021: (Start)
a(n)-a(n-2) = A008680(n).
a(n)-a(n-3) = A025802(n).
a(n)-a(n-4) = A025795(n).
a(n)-a(n-5) = A005044(n+3). (End)
a(n)= floor((n^3 + 21*n^2 + 156*n - 45*n*(n mod 2) + 720)/720 - [(n mod 10)=1]/5). - Hoang Xuan Thanh, Aug 20 2025

A001996 Number of partitions of n into parts 2, 3, 4, 5, 6, 7.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 17, 23, 26, 33, 37, 47, 52, 64, 72, 86, 96, 115, 127, 149, 166, 192, 212, 245, 269, 307, 338, 382, 419, 472, 515, 576, 629, 699, 760, 843, 913, 1007, 1091, 1197, 1293, 1416, 1525, 1663, 1790, 1945, 2088, 2265, 2426

Offset: 0

N. J. A. Sloane

Also, Molien series for invariants of finite Coxeter group A_6. The Molien series for the finite Coxeter group of type A_k (k >= 1) has G.f. = 1/Prod_{i=2..k+1} (1-x^i). - N. J. A. Sloane, Jan 11 2016

Cayley tabulates the coefficients in the expansion of H = 1 / ((1 - x^2) * (1 - x^4) * ... * (1 - x^14)) with even indices 0, 2, ..., 142.

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
G.f. = 1 + q^2 + q^6 + 2*q^8 + 2*q^10 + 4*q^12 + 4*q^14 + 6*q^16 + ...

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.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
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
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, American Journal of Mathematics, 2 (1879), pp.71-84. See pp.77-78.
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]
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -2, -2, -1, 0, 2, 2, 2, 2, 0, -1, -2, -2, -1, 0, 0, 1, 1, 1, 0, -1).

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

nn = 102; t = CoefficientList[Series[1/((1 - x^4)*(1 - x^6)*(1 - x^8)*(1 - x^10)*(1 - x^12)*(1 - x^14)), {x, 0, nn}], x]; t = Take[t, {1, nn, 2}]

G.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).

Euler transform of length 7 sequence [ 0, 1, 1, 1, 1, 1, 1]. - Michael Somos, Apr 23 2014

More terms from James Sellers, Feb 09 2000

A037145 Expansion of 1/((1-x^2)(1-x^3)...(1-x^6)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 6, 6, 9, 9, 14, 13, 19, 20, 26, 27, 36, 36, 47, 49, 60, 63, 78, 80, 97, 102, 120, 126, 149, 154, 180, 189, 216, 227, 260, 270, 307, 322, 361, 378, 424, 441, 492, 515, 568, 594, 656, 682, 750

Offset: 0

N. J. A. Sloane

Also, Molien series for invariants of finite Coxeter group A_5. The Molien series for the finite Coxeter group of type A_k (k >= 1) has G.f. = 1/Prod_{i=2..k+1} (1-x^i). - N. J. A. Sloane, Jan 11 2016

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

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

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

Cf. A001402 (partial sums).

CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,6}],{x,0,50}],x] (* Harvey P. Dale, Dec 25 2012 *)

A266776 Molien series for invariants of finite Coxeter group A_7.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 7, 11, 12, 18, 19, 27, 30, 40, 44, 58, 64, 82, 91, 113, 126, 155, 171, 207, 230, 274, 303, 358, 395, 462, 509, 589, 649, 746, 818, 934, 1024, 1161, 1269, 1432, 1562, 1753, 1909, 2131, 2317, 2577, 2794, 3095, 3352, 3698, 3997, 4396, 4743, 5200, 5601, 6121, 6584, 7177, 7705, 8377, 8983, 9741, 10429, 11285, 12065

Offset: 0

N. J. A. Sloane, Jan 11 2016

The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1 - x^i).

Note that this is the root system A_k, not the alternating group Alt_k.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -1, -2, -2, -1, 1, 2, 2, 3, 2, 1, -1, -2, -3, -2, -2, -1, 1, 2, 2, 1, 1, 0, 0, -1, -1, -1, 0, 1).
Index entries for Molien series

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(&*[1-t^k: k in [2..8]]))); // G. C. Greubel, Oct 24 2018

CoefficientList[Series[1/Product[1-t^k, {k,2,8}], {t, 0, 40}], t] (* G. C. Greubel, Oct 24 2018 *)

t='t+O('t^40); Vec(1/prod(k=2,8, 1-t^k)) \\ G. C. Greubel, Oct 24 2018

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)).

A266781 Molien series for invariants of finite Coxeter group A_12.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 33, 40, 53, 63, 83, 98, 126, 150, 188, 223, 278, 327, 401, 473, 573, 672, 809, 944, 1126, 1312, 1551, 1800, 2118, 2446, 2859, 3295, 3829, 4395, 5086, 5817, 6699, 7642, 8760, 9961, 11380, 12898, 14678, 16596, 18819, 21217, 23987, 26971, 30397, 34099, 38316, 42877, 48058, 53649, 59972, 66811

Offset: 0

N. J. A. Sloane, Jan 11 2016

The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1 - x^i).

Note that this is the root system A_k, not the alternating group Alt_k.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, 0, 1, 2, 3, 3, 3, 2, 0, -1, -2, -3, -4, -4, -5, -4, -3, -1, 1, 3, 5, 7, 7, 6, 5, 3, 2, -1, -4, -6, -7, -8, -7, -6, -4, -1, 2, 3, 5, 6, 7, 7, 5, 3, 1, -1, -3, -4, -5, -4, -4, -3, -2, -1, 0, 2, 3, 3, 3, 2, 1, 0, -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, 1, 1, 1, 0, -1).
Index entries for Molien series

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^j: j in [2..13]]) )); // G. C. Greubel, Feb 04 2020

S:=series(1/mul(1-x^j, j=2..13)), x, 75):
seq(coeff(S, x, j), j=0..70); # G. C. Greubel, Feb 04 2020

CoefficientList[Series[1/Product[1-x^j, {j,2,13}], {x,0,70}], x] (* G. C. Greubel, Feb 04 2020 *)
LinearRecurrence[{0,1,1,1,0,0,-1,-1,-1,-1,-1,0,0,-1,0,1,2,3,3,3,2,0,-1,-2,-3,-4,-4,-5,-4,-3,-1,1,3,5,7,7,6,5,3,2,-1,-4,-6,-7,-8,-7,-6,-4,-1,2,3,5,6,7,7,5,3,1,-1,-3,-4,-5,-4,-4,-3,-2,-1,0,2,3,3,3,2,1,0,-1,0,0,-1,-1,-1,-1,-1,0,0,1,1,1,0,-1},{1,0,1,1,2,2,4,4,7,8,12,14,21,24,33,40,53,63,83,98,126,150,188,223,278,327,401,473,573,672,809,944,1126,1312,1551,1800,2118,2446,2859,3295,3829,4395,5086,5817,6699,7642,8760,9961,11380,12898,14678,16596,18819,21217,23987,26971,30397,34099,38316,42877,48058,53649,59972,66811,74499,82813,92136,102204,113455,125613,139140,153754,169979,187481,206857,227767,250835,275713,303108,332617,365036,399950,438201,479372,524403,572813,625657,682451,744307,810735},80] (* Harvey P. Dale, Jul 01 2021 *)

Vec( 1/prod(j=2,13, 1-x^j) +O('x^70) ) \\ G. C. Greubel, Feb 04 2020

def A266781_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/prod(1-x^j for j in (2..13)) ).list()
A266781_list(70) # G. C. Greubel, Feb 04 2020

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)*(1-t^9)*(1-t^10)*(1-t^11)*(1-t^12)*(1-t^13)).

A370256 The number of ways in which n can be expressed as b^2 * c^3, with b and c >= 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Feb 23 2024

First differs from A075802 and A112526 at n = 64.
The least number k such that a(k) = n is A005179(n)^6.
The indices of records are the sixth powers of the highly composite numbers, A002182(n)^6.

			1 = 1^2 * 1^3, so a(1) = 1.
64 = 1^2 * 4^3 = 8^2 * 1^3, so a(64) = 2.
4096 = 64^2 * 1^3 = 8^2 * 4^3 = 1^2 * 16^3, so a(4096)= 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001694, A002182, A005179, A057523, A075802, A078434, A103221, A112526.

f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> (x+2)\2 - (x+2)\3, factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, 1/((1 - X^2)*(1 - X^3)))[n], ", ")) \\ Vaclav Kotesovec, Feb 23 2024

from math import prod
from sympy import factorint
def A370256(n): return prod((e>>1)+1-(e+2)//3 for e in factorint(n).values()) # Chai Wah Wu, Apr 15 2025

Multiplicative with a(p^e) = A103221(e).
a(n) > 0 if and only if n is a powerful number (A001694).
a(A001694(n)) = A057523(n).
a(n^6) = A000005(n).
Sum_{k=1..n} a(k) ~ zeta(3/2) * sqrt(n) + zeta(2/3) * n^(1/3).
Dirichlet generating function: zeta(2*s)*zeta(3*s). - Vaclav Kotesovec, Feb 23 2024

cp's OEIS Frontend

A128115 Mobius inversion of A103221.

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A016921 a(n) = 6*n + 1.

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A008615 a(n) = floor(n/2) - floor(n/3).

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A266755 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^4)).

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

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A266755 Expansion of 1/((1-x^2)(1-x^3)(1-x^4)).

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