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A069905
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Number of partitions of n into 3 positive parts.
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13
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0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208, 217, 225, 234, 243, 252, 261, 271, 280, 290, 300, 310, 320, 331, 341
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OFFSET
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0,6
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COMMENTS
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Number of binary bracelets of n beads, 3 of them 0. For n>=3 a(n-3) is the number of binary bracelets of n beads, 3 of them 0, with 00 prohibited. [Washington Bomfim, Aug 27 2008]
Also number of partitions of n-3 into parts 1, 2, and 3. [Joerg Arndt, Sep 05 2013]
a(n) = a(n-1) + A008615(n-1) for n > 0. - Reinhard Zumkeller, Apr 28 2014
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410.
D. E. Knuth, The Art of Computer Programming, vol. 4,fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.
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LINKS
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Washington Bomfim, Table of n, a(n) for n = 0..10000
Index to sequences with linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).
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FORMULA
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G.f.: x^3/((1-x)*(1-x^2)*(1-x^3)).
a(n) = round(n^2/12).
a(n) = floor((n^2+6)/12). - Washington Bomfim, Jul 03 2012
a(-n) = a(n). - Michael Somos, Sep 04 2013
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EXAMPLE
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x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 8*x^10 + 10*x^11 + ...
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MAPLE
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A069905 := n->round(n^2/12);
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MATHEMATICA
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a[ n_] := Round[ n^2 / 12] (* Michael Somos, Sep 04 2013 *)
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PROG
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(PARI) a(n) = floor((n^2+6)/12); /* Washington Bomfim, Jul 03 2012 */
(Haskell)
a069905 n = a069905_list !! n
a069905_list = scanl (+) 0 a008615_list
-- Reinhard Zumkeller, Apr 28 2014
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CROSSREFS
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Another version of A001399, which is the main entry for this sequence.
Sequence in context: A034092 A211540 A001399 * A008761 A008760 A008759
Adjacent sequences: A069902 A069903 A069904 * A069906 A069907 A069908
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, May 04 2002
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STATUS
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approved
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