Search: id:A007690
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%I A007690 M0167
%S A007690 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 A007690 199,193,275,263,385,364,516,511,694,686,946,925,1246,1260,1650,1663,
%U A007690 2194,2202,2857,2928,3721,3813,4866,4967,6257,6487,8051,8342,10369
%N A007690 Number of partitions of n in which no part occurs just once.
%C A007690 Euler transform of period 6 sequence [0,1,1,1,0,1,...]. - Michael Somos 
               Apr 21 2004
%C A007690 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 (deutsch(AT)duke.poly.edu), 
               Feb 16 2006
%C A007690 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 A007690 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] * ... [From Mats 
               Granvik (mats.granvik(AT)abo.fi), Aug 07 2009]
%D A007690 G. E. Andrews, Number Theory, Dover Publications, 1994. page 197. MR1298627
%D A007690 R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 242.
%D A007690 George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, 
               Mass., 1976, p. 14, Example 9.
%D A007690 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (p. 14, 
               Example 9).
%D A007690 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 A007690 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A007690 A. E. Holroyd, T. M. Liggett and D. Romik, <a href="http://arXiv.org/
               abs/math.PR/0302216">Integrals, partitions and cellular automata</
               a>
%H A007690 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PartitionFunctionP.html">Partition Function P</a>
%F A007690 G.f.: Prod{k>0 is a multiple of 2 or 3} (1/(1-x^k)). - Christian G. Bower 
               (bowerc(AT)usa.net), Jun 23 2000
%F A007690 G.f.: product{i=1, oo, (1+x^3j)/(1-x^2j)} - Jon Perry (perry(AT)globalnet.co.uk), 
               Mar 29 2004
%F A007690 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 g.f. for 
               A137566. - Michael Somos, Jan 26 2008
%e A007690 a(6)=4 because we have [3,3],[2,2,2],[2,2,1,1] and [1,1,1,1,1,1].
%e A007690 q + q^49 + q^73 + 2*q^97 + q^121 + 4*q^145 + 2*q^169 + 6*q^193 + ...
%e A007690 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 
               + ...
%p A007690 G:=product((1-x^j+x^(2*j))/(1-x^j),j=1..70): Gser:=series(G,x=0,60): 
               1, seq(coeff(Gser,x^n),n=1..54); - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Feb 10 2006
%o A007690 (PARI) a(n)=local(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 A007690 Cf. A000041, A055922, A055923, A114917, A114918.
%Y A007690 Sequence in context: A154280 A004795 A161268 this_sequence A143375 A074364 
               A008796
%Y A007690 Adjacent sequences: A007687 A007688 A007689 this_sequence A007691 A007692 
               A007693
%K A007690 nonn
%O A007690 0,5
%A A007690 N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)

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