A222591 Numerators of (n*(n - 3)/6) + 1, arising as the maximum possible number of triple lines for an n-element set.

1, 5, 8, 4, 17, 23, 10, 38, 47, 19, 68, 80, 31, 107, 122, 46, 155, 173, 64, 212, 233, 85, 278, 302, 109, 353, 380, 136, 437, 467, 166, 530, 563, 199, 632, 668, 235, 743, 782, 274, 863, 905, 316, 992, 1037, 361, 1130, 1178, 409, 1277, 1328

Offset

3,2

Comments

Numerators of (n*(n - 3)/6) + 1, which arises as the maximum possible number of triple lines for an n-element set, according to Green and Tao, cited in Elekes. The fractions for n = 3, 4, 5, 6, ... are 1/1, 5/3, 8/3, 4/1, 17/3, 23/3, 10/1, 38/3, 47/3, 19/1, 68/3, 80/3, 31/1, 107/3, 122/3, 46/1, 155/3, 173/3, 64/1, 212/3, 233/3, 85/1, 278/3, 302/3, 109/1, 353/3, 380/3, 136/1, 437/3, 467/3, 166/1, 530/3, 563/3, 199/1, 632/3, 668/3, 235/1, 743/3, 782/3, 274/1, 863/3, 905/3, 316/1, 992/3, 1037/3, 361/1, 1130/3, 1178/3, 409/1, 1277/3, 1328/3. The corresponding denominators are A169609.

Links

Table of n, a(n) for n=3..53.

György Elekes, Endre Szabó, On Triple Lines and Cubic Curves --- the Orchard Problem revisited, arXiv:1302.5777 [math.CO], Feb 23, 2013.

Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Example

a(10) = 38 because (10*(10 - 3)/6) + 1 = 38/3.

Mathematica

Numerator[Table[(n(n-3))/6+1, {n, 3, 60}]] (* or *) LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 5, 8, 4, 17, 23, 10, 38, 47}, 60] (* Harvey P. Dale, Feb 11 2015 *)

Crossrefs

Cf. A169609.

Sequence in context: A020857 A096413 A334116 * A299447 A300085 A186691

Adjacent sequences: A222588 A222589 A222590 * A222592 A222593 A222594

Keyword

nonn,easy,frac

Author

Jonathan Vos Post, Feb 25 2013

Status

approved