A128422 Projective plane crossing number of K_{4,n}.

0, 0, 0, 2, 4, 6, 10, 14, 18, 24, 30, 36, 44, 52, 60, 70, 80, 90, 102, 114, 126, 140, 154, 168, 184, 200, 216, 234, 252, 270, 290, 310, 330, 352, 374, 396, 420, 444, 468, 494, 520, 546, 574, 602, 630, 660, 690, 720, 752, 784, 816, 850, 884, 918, 954, 990, 1026

Offset: 1

Author

Eric W. Weisstein, Mar 02 2007

Keywords

• nonn
• easy

Comments

From Gus Wiseman, Oct 15 2020: (Start)

Also the number of 3-part compositions of n that are neither strictly increasing nor weakly decreasing. The set of numbers k such that row k of A066099 is such a composition is the complement of A333255 (strictly increasing) and A114994 (weakly decreasing) in A014311 (triples). The a(4) = 2 through a(9) = 14 compositions are:

(1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6) (1,1,7)

(1,2,1) (1,2,2) (1,3,2) (1,3,3) (1,4,3) (1,4,4)

(1,3,1) (1,4,1) (1,4,2) (1,5,2) (1,5,3)

(2,1,2) (2,1,3) (1,5,1) (1,6,1) (1,6,2)

(2,3,1) (2,1,4) (2,1,5) (1,7,1)

(3,1,2) (2,2,3) (2,2,4) (2,1,6)

(2,3,2) (2,3,3) (2,2,5)

(2,4,1) (2,4,2) (2,4,3)

(3,1,3) (2,5,1) (2,5,2)

(4,1,2) (3,1,4) (2,6,1)

(3,2,3) (3,1,5)

(3,4,1) (3,2,4)

(4,1,3) (3,4,2)

(5,1,2) (3,5,1)

(4,1,4)

(4,2,3)

(5,1,3)

(6,1,2)

(End)

Links

• Eric Weisstein's World of Mathematics, Complete Bipartite Graph
• Eric Weisstein's World of Mathematics, Projective Plane Crossing Number
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

A007997 counts the complement.

A337482 counts these compositions of any length.

A337484 is the non-strict/non-strict version.

A000009 counts strictly increasing compositions, ranked by A333255.

A000041 counts weakly decreasing compositions, ranked by A114994.

A001523 counts unimodal compositions (strict: A072706).

A007318 and A097805 count compositions by length.

A032020 counts strict compositions, ranked by A233564.

A225620 ranks weakly increasing compositions.

A333149 counts neither increasing nor decreasing strict compositions.

A333256 ranks strictly decreasing compositions.

A337483 counts 3-part weakly increasing or weakly decreasing compositions.

Cf. A000217, A001399, A001840, A059204, A072705, A115981, A329398, A332578, A332831, A332833, A332834, A332874, A333147, A333190.

Programs

Mathematica

Table[Floor[((n - 2)^2 + (n - 2))/3], {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)
Table[Ceiling[n^2/3] - n, {n, 20}] (* Eric W. Weisstein, Sep 07 2018 *)
Table[(3 n^2 - 9 n + 4 - 4 Cos[2 n Pi/3])/9, {n, 20}] (* Eric W. Weisstein, Sep 07 2018 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 0, 0, 2, 4, 6}, 20] (* Eric W. Weisstein, Sep 07 2018 *)
CoefficientList[Series[-2 x^3/((-1 + x)^3 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 07 2018 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&&!GreaterEqual@@#&]],{n,15}] (* Gus Wiseman, Oct 15 2020 *)

PARI

a(n)=(n-1)*(n-2)\3 \\ Charles R Greathouse IV, Jun 06 2013

Formula

a(n) = floor(n/3)*(2n-3(floor(n/3)+1)).

a(n) = ceiling(n^2/3) - n. - Charles R Greathouse IV, Jun 06 2013

G.f.: -2*x^4 / ((x-1)^3*(x^2+x+1)). - Colin Barker, Jun 06 2013

a(n) = floor((n - 1)(n - 2) / 3). - Christopher Hunt Gribble, Oct 13 2009

a(n) = 2*A001840(n-3). - R. J. Mathar, Jul 21 2015

a(n) = A000217(n-2) - A001399(n-6) - A001399(n-3). - Gus Wiseman, Oct 15 2020

Sum_{n>=4} 1/a(n) = 10/3 - Pi/sqrt(3). - Amiram Eldar, Sep 27 2022