A014591 a(n) = floor(n^2/12 + 5/4).

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, 57, 62, 66, 71, 76, 81, 86, 92, 97, 103, 109, 115, 121, 128, 134, 141, 148, 155, 162, 170, 177, 185, 193, 201, 209, 218, 226, 235, 244, 253, 262, 272, 281, 291, 301, 311, 321

Offset: 0

Author

David Broadhurst

Keywords

• nonn
• easy

Comments

Number of partitions of n + 10 into 4 distinct parts one of which is 3. - Michael Somos, Nov 03 2011

Number of partitions of n into 3 or fewer distinct parts. - Mo Li, Sep 27 2019

Examples

			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 + ...
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.
		

Links

• Jan Kneissler, The number of primitive Vassiliev invariants of degree up to 12, arXiv:q-alg/9706022, 1997.

Crossrefs

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.

Programs

Mathematica

Floor[Range[0,70]^2/12+5/4] (* Harvey P. Dale, Oct 22 2013 *)
Table[Length[Select[IntegerPartitions[k, 3], DuplicateFreeQ]], {k,1,50}] (* Mo Li, Sep 27 2019 *)

PARI

{a(n) = (n^2 + 3) \ 12 + 1} /* Michael Somos, Nov 03 2011 */

Formula

G.f.: (1/(1-x^3)-x^2)/(1-x)/(1-x^2).

a(-n) = a(n). a(n) = 1 + A069905(n). - Michael Somos, Nov 03 2011

Extensions

More terms from Erich Friedman