A193802 -id:A193802 - cp's OEIS frontend

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

Hugo Pfoertner, Jul 12 2017

For definitions see A103294.

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.

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

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

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

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

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)

A193803 Length of perfect Wichmann rulers.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 22, 29, 36, 43, 46, 50, 57, 64, 68, 71, 79, 90, 101, 108, 112, 123, 134, 138, 145, 153, 156, 168, 175, 183

Offset: 1

Peter Luschny, Oct 22 2011

R is a perfect Wichmann ruler iff R is a perfect ruler (for definition see A103294) and there exist two integers r>=0 and s>=0 such that the type of the difference representation of the ruler is [1*r, r+1, (2r+1)*r, (4r+3)*s, (2r+2)*(r+1), 1*r].

			[0, 1, 2, 5, 10, 15, 26, 37, 48, 54, 60, 66, 67, 68] is a perfect Wichmann ruler with length 68 of Wichmann type (2,3). By contrast [0, 1, 2, 8, 15, 16, 26, 36, 46, 56, 59, 63, 65, 68] is a perfect ruler with length 68 which is not a Wichmann ruler.

L. Egidi and G. Manzini, Spaced seeds design using perfect rulers, Tech. Rep. CS Department Universita del Piemonte Orientale, June 2011.
Peter Luschny, Perfect rulers
B. Wichmann, A note on restricted difference bases, J. Lond. Math. Soc. 38 (1963), 465-466.

Cf. A004137, A193802.

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

Original entry on oeis.org

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

Hugo Pfoertner, Jul 14 2017

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

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

Cf. A004137, A193802, A193803, A289761.

A014641 is a subsequence.

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

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

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

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

cp's OEIS Frontend

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

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A193803 Length of perfect Wichmann rulers.

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

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