A289761 Maximum length of a perfect Wichmann ruler with n segments.

3, 6, 9, 12, 15, 22, 29, 36, 43, 50, 57, 68, 79, 90, 101, 112, 123, 138, 153, 168, 183, 198, 213, 232, 251, 270, 289, 308, 327, 350, 373, 396, 419, 442, 465, 492, 519, 546, 573, 600, 627, 658, 689, 720, 751, 782, 813, 848, 883, 918, 953, 988, 1023, 1062, 1101, 1140, 1179, 1218, 1257, 1300, 1343, 1386, 1429

Offset: 2

Author

Hugo Pfoertner, Jul 12 2017

Keywords

• nonn
• easy

Comments

For definitions see A103294.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 2..10001
• Peter Luschny, Are optimal rulers of Wichmann type?
• Eric Weisstein's World of Mathematics, Dark Satanic Mills on a Cloudy Day.
• Eric Weisstein's World of Mathematics, Sparse Ruler.
• Eric Weisstein's World of Mathematics, Wichmann Ruler.
• B. Wichmann, A note on restricted difference bases, J. Lond. Math. Soc. 38 (1963), 465-466.

Crossrefs

Cf. A004137, A103294, A193802, A193803, A289873.

Programs

Mathematica

Table[(n^2 - (Mod[n, 6] - 3)^2)/3 + n, {n, 2, 66}] (* Michael De Vlieger, Jul 14 2017 *)

PARI

a(n) = n + (n^2 - (n%6 - 3)^2)/3; \\ Michel Marcus, Jul 14 2017

Python

def A289761(n): return (n+(m:=n%6))*(n-(k:=m-3))//3+k # Chai Wah Wu, Jun 20 2024

Formula

a(n) = ( n^2 - (mod(n,6)-3)^2 ) / 3 + n.

Conjectures from Colin Barker, Jul 14 2017: (Start)

G.f.: x^2*(3 + 4*x^5 - 3*x^6) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).

a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8) for n>9.

(End)