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A128422
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Projective plane crossing number of K_{4,n}.
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11
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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
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OFFSET
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1,4
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COMMENTS
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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)
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LINKS
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FORMULA
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a(n) = floor(n/3)*(2n-3(floor(n/3)+1)).
G.f.: -2*x^4 / ((x-1)^3*(x^2+x+1)). - Colin Barker, Jun 06 2013
Sum_{n>=4} 1/a(n) = 10/3 - Pi/sqrt(3). - Amiram Eldar, Sep 27 2022
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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A337482 counts these compositions of any length.
A337484 is the non-strict/non-strict version.
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.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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