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A002133
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Number of partitions of n using only 2 types of parts.
(Formerly M1324 N0507)
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9
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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; internal format)
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OFFSET
| 1,4
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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.
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REFERENCES
| Andrews, George E., 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.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
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).
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LINKS
| 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).
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FORMULA
| G.f.=sum(sum(x^(i+j)/[(1-x^i)(1-x^j)], j=1..i-1), i=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
Andrews gives a formula which 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 (vladeta(AT)eunet.rs), Sep 18 2007
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EXAMPLE
| a(8)=13 because we have 71, 62, 611, 53, 5111, 422, 41111, 332, 3311, 311111, 22211, 221111, 2111111.
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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 (deutsch(AT)duke.poly.edu), 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
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CROSSREFS
| A diagonal of A060177.
Cf. A002134.
Sequence in context: A030130 A164874 A045845 * A092306 A090552 A024520
Adjacent sequences: A002130 A002131 A002132 * A002134 A002135 A002136
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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