A289873 Related to perfect Wichmann rulers: a(n) = ( n^2 - (mod(n, 6) - 3)^2 ) / 3.

1, 3, 5, 7, 9, 15, 21, 27, 33, 39, 45, 55, 65, 75, 85, 95, 105, 119, 133, 147, 161, 175, 189, 207, 225, 243, 261, 279, 297, 319, 341, 363, 385, 407, 429, 455, 481, 507, 533, 559, 585, 615, 645, 675, 705, 735, 765, 799, 833, 867, 901, 935, 969, 1007, 1045, 1083, 1121, 1159, 1197, 1239, 1281, 1323, 1365

Offset: 2

Author

Hugo Pfoertner, Jul 14 2017

Keywords

• nonn
• easy

Comments

Leading term in length A289761 of longest perfect Wichmann ruler with n segments.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 2..10001

Crossrefs

Cf. A004137, A193802, A193803, A289761.

A014641 is a subsequence.

Programs

Maple

p := (n, x) -> (2*n - 3*(1 + x))*(1 + x):
a := n -> p(n, 2*floor(n/6)):
seq(a(n), n = 2..64); # Peter Luschny, Jul 14 2017

Mathematica

Table[(n^2 - (Mod[n, 6] - 3)^2)/3, {n, 2, 64}] (* Michael De Vlieger, Jul 14 2017 *)

Python

def A289873(n): return (n+(m:=n%6))*(n-(k:=m-3))//3+k-n # Chai Wah Wu, Jun 20 2024

Formula

a(n) = A289761(n) - n.

G.f.: x^2*(1 + x - x^2)*(1 + x^2 - x^3 + 2*x^4 + x^5) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) (conjectured). - Colin Barker, Jul 14 2017

Can be seen as a family of parabolas p_{n}(x) = (2*n - 3*(1 + x))*(1 + x) evaluated at x = 2*floor(n/6). - Peter Luschny, Jul 14 2017