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 A002133 Number of partitions of n with exactly two part sizes. (Formerly M1324 N0507) 21
 0, 0, 1, 2, 5, 6, 11, 13, 17, 22, 27, 29, 37, 44, 44, 55, 59, 68, 71, 81, 82, 102, 97, 112, 109, 136, 126, 149, 141, 168, 157, 188, 176, 212, 182, 231, 207, 254, 230, 266, 241, 300, 259, 319, 283, 344, 295, 373, 311, 386, 352, 417, 353, 452, 368, 460, 418, 492, 413 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Also number of solutions to the Diophantine equation ab + bc + cd = n, with a,b,c >= 1. - N. J. A. Sloane, Jun 17 2011 A generalized sum of divisors function. REFERENCES 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). LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_3(n). E. T. Bell, The form wx+xy+yz+zu, Bull. Amer. Math. Soc., 42 (1936), 377-380. D. Christopher, T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), #15.11.5. W. J. Keith, Partitions into a small number of part sizes, Int. Jour. of Num. Thy., Vol 13 no. 1, 229-241 (2017), doi:10.1142/S1793042117500130 P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341. N. Benyahia Tani, S. Bouroubi, O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Bulletin du Laboratoire, 03 (2015) 18-27. N. B. Tani and S. Bouroubi, Enumeration of the Partitions of an Integer into Parts of a Specified Number of Different Sizes and Especially Two Sizes, J. Integer Seqs., Vol. 14 (2011), #11.3.6. (This sequence appears as the rightmost column of Table 1 on p. 10.) FORMULA G.f.: Sum_{i>=1} Sum_{j=1..i-1} x^(i+j)/((1-x^i)*(1-x^j)). - Emeric Deutsch, Mar 30 2006 Andrews gives a formula which is programmed up in the Maple code below. - N. J. A. Sloane, Jun 17 2011 G.f.: (G(x)^2-H(x))/2 where G(x) = Sum_{k>0} x^k/(1-x^k) and H(x) = Sum_{k>0} x^(2*k)/(1-x^k)^2. More generally, we obtain g.f. for number of partitions of n with m types of parts if we substitute x(i) with -Sum_{k>0}(x^n/(x^n-1))^i in cycle index Z(S(m); x(1),x(2),...,x(m)) of symmetric group S(m) of degree m. - Vladeta Jovovic, Sep 18 2007 EXAMPLE a(8) = 13 because we have 71, 62, 611, 53, 5111, 422, 41111, 332, 3311, 311111, 22211, 221111, 2111111. MAPLE g:=sum(sum(x^(i+j)/(1-x^i)/(1-x^j), j=1..i-1), i=1..80): gser:=series(g, x=0, 65): seq(coeff(gser, x^n), n=1..60); # Emeric Deutsch, Mar 30 2006 with(numtheory); D00:=n->add(tau(j)*tau(n-j), j=1..n-1); L3:=n->(D00(n)+tau(n)-sigma(n))/2; [seq(L3(n), n=1..60)]; # N. J. A. Sloane, Jun 17 2011 MATHEMATICA nn=50; ss=Sum[Sum[x^(i+j)/(1-x^i)/(1-x^j), {j, 1, i-1}], {i, 1, nn}]; Drop[CoefficientList[Series[ss, {x, 0, nn}], x], 1]  (* Geoffrey Critzer, Sep 13 2012 *) CROSSREFS A diagonal of A060177. Cf. A002134. Sequence in context: A164874 A337498 A045845 * A092306 A319242 A323398 Adjacent sequences:  A002130 A002131 A002132 * A002134 A002135 A002136 KEYWORD nonn,look,easy AUTHOR STATUS approved

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Last modified April 18 03:59 EDT 2021. Contains 343072 sequences. (Running on oeis4.)