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%I #26 Sep 27 2019 10:03:47
%S 1,1,1,2,2,3,4,5,6,8,9,11,13,15,17,20,22,25,28,31,34,38,41,45,49,53,
%T 57,62,66,71,76,81,86,92,97,103,109,115,121,128,134,141,148,155,162,
%U 170,177,185,193,201,209,218,226,235,244,253,262,272,281,291,301,311,321
%N a(n) = floor(n^2/12 + 5/4).
%C Number of partitions of n + 10 into 4 distinct parts one of which is 3. - _Michael Somos_, Nov 03 2011
%C Number of partitions of n into 3 or fewer distinct parts. - _Mo Li_, Sep 27 2019
%H Jan Kneissler, <a href="https://arxiv.org/abs/q-alg/9706022">The number of primitive Vassiliev invariants of degree up to 12</a>, arXiv:q-alg/9706022, 1997.
%F G.f.: (1/(1-x^3)-x^2)/(1-x)/(1-x^2).
%F a(-n) = a(n). a(n) = 1 + A069905(n). - _Michael Somos_, Nov 03 2011
%e 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 8*x^9 + ...
%e 10 = 4 + 3 + 2 + 1, 11 = 5 + 3 + 2 + 1, 12 = 6 + 3 + 2 + 1, 13 = 7 + 3 + 2 + 1 = 5 + 4 + 3 + 1, 14 = 8 + 3 + 2 + 1 = 5 + 4 + 3 + 2, 15 = 9 + 3 + 2 + 1 = 6 + 5 + 3 + 1 = 6 + 4 + 3 + 2.
%t Floor[Range[0,70]^2/12+5/4] (* _Harvey P. Dale_, Oct 22 2013 *)
%t Table[Length[Select[IntegerPartitions[k, 3], DuplicateFreeQ]], {k,1,50}] (* _Mo Li_, Sep 27 2019 *)
%o (PARI) {a(n) = (n^2 + 3) \ 12 + 1} /* _Michael Somos_, Nov 03 2011 */
%Y It may be only a coincidence that the first 11 terms reproduce all available data on Vassiliev invariants from diagrams with u=2 univalent vertices, as recorded in the Kneissler paper.
%K nonn,easy
%O 0,4
%A _David Broadhurst_
%E More terms from _Erich Friedman_
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