%I M0167 #56 Aug 17 2020 22:14:47

%S 1,0,1,1,2,1,4,2,6,5,9,7,16,11,22,20,33,28,51,42,71,66,100,92,147,131,

%T 199,193,275,263,385,364,516,511,694,686,946,925,1246,1260,1650,1663,

%U 2194,2202,2857,2928,3721,3813,4866,4967,6257,6487,8051,8342,10369

%N Number of partitions of n in which no part occurs just once.

%C Also number of partitions of n into parts, each larger than 1, such that consecutive integers do not both appear as parts. Example: a(6)=4 because we have [6], [4,2], [3,3] and [2,2,2]. - _Emeric Deutsch_, Feb 16 2006

%C Also number of partitions of n into parts divisible by 2 or 3. - Alexander E. Holroyd (holroyd(AT)math.ubc.ca), May 28 2008

%C Infinite convolution product of [1,0,1,1,1,1,1] aerated n-1 times. i.e. [1,0,1,1,1,1,1] * [1,0,0,0,1,0,1] * [1,0,0,0,0,0,1] * ... . - _Mats Granvik_, Aug 07 2009

%D G. E. Andrews, Number Theory, Dover Publications, 1994. page 197. MR1298627

%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976, p. 14, Example 9.

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.6).

%D R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 242.

%D P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 54, Article 300.

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

%H Alois P. Heinz, <a href="/A007690/b007690.txt">Table of n, a(n) for n = 0..10000</a>

%H A. E. Holroyd, T. M. Liggett and D. Romik, <a href="https://arxiv.org/abs/math/0302216">Integrals, partitions and cellular automata</a>, arXiv:math/0302216 [math.PR], 2003.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>

%F G.f.: Product_{k>0 is a multiple of 2 or 3} (1/(1-x^k)). - _Christian G. Bower_, Jun 23 2000

%F G.f.: Product_{j>=1} (1+x^(3*j)) / (1-x^(2*j)). - _Jon Perry_, Mar 29 2004

%F Euler transform of period 6 sequence [0, 1, 1, 1, 0, 1, ...]. - _Michael Somos_, Apr 21 2004

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (864 t)) = 1/6 (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137566. - _Michael Somos_, Jan 26 2008

%F From _Alois P. Heinz_, Oct 09 2011: (Start)

%F a(n) = A000041(n) - A183558(n).

%F a(n) = A183568(n,0) - A183568(n,1).

%F G.f.: Product_{j>0} (1-x^j+x^(2*j)) / (1-x^j). (End)

%F a(n) ~ exp(2*Pi*sqrt(n)/3)/(6*sqrt(2)*n). - _Vaclav Kotesovec_, Sep 23 2015

%F a(n) = A000009(n/3) - Sum_{k>=1} (-1)^k a(n - k*(3*k +/- 1)). - _Peter J. Taylor_, May 16 2019

%e a(6) = 4 because we have [3,3], [2,2,2], [2,2,1,1] and [1,1,1,1,1,1].

%e G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 6*x^8 + 5*x^9 + 9*x^10 + ...

%e G.f. = q + q^49 + q^73 + 2*q^97 + q^121 + 4*q^145 + 2*q^169 + 6*q^193 + ...

%p G:= mul((1-x^j+x^(2*j))/(1-x^j), j=1..70): Gser:=series(G, x, 60): seq(coeff(Gser, x, n), n=0..54); # _Emeric Deutsch_, Feb 10 2006

%t nn=40;CoefficientList[Series[Product[1/(1-x^i)-x^i,{i,1,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Dec 02 2012 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^6] / (QPochhammer[ x^2] QPochhammer[ x^3]), {x, 0, n}]; (* _Michael Somos_, Feb 22 2015 *)

%t nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k))/(1 - x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 23 2015 *)

%t Table[Length@Select[Tally /@ IntegerPartitions@n, AllTrue[#, Last[#] > 1 &] &], {n, 0, 54}] (* _Robert Price_, Aug 17 2020 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* _Michael Somos_, Apr 21 2004 */

%Y Cf. A000041, A055922, A055923, A114917, A114918, A183558, A183568.

%Y Cf. A100405, A160974-A160990.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, _Robert G. Wilson v_

%E Minor edits by _Vaclav Kotesovec_, Aug 23 2015