cp's OEIS Frontend

a(n-2) appears to be the minimum number of rectangular tiles to place on an n X n grid of unit squares, possibly of different sizes, such that each side of every tile lies on a grid line, every unit square is covered by at most one tile, and each row and each column of the grid has exactly one unit square that is not covered by any tile. - Yifan Xie, Jul 19 2025 [The conjecture is proven. - Yifan Xie, Jul 24 2025]

a(n) is the least integer such that there is an n-element set of integers between 0 and a(n), the sums of pairs (of not necessarily distinct elements) of which are distinct.

From David W. Wilson, Aug 17 2007: (Start)

An n-mark Golomb ruler has a unique integer distance between any pair of marks and thus measures n(n-1)/2 distinct integer distances.

An optimal n-mark Golomb ruler has the smallest possible length (distance between the two end marks) for an n-mark ruler.

A perfect n-mark Golomb ruler has length exactly n(n-1)/2 and measures each distance from 1 to n(n-1)/2. (End)

Positions where A143824 increases (see also A227590). - N. J. A. Sloane, Apr 08 2016

From Gus Wiseman, May 17 2019: (Start)

Also the smallest m such that there exists a length-n composition of m for which every restriction to a subinterval has a different sum. Representatives of compositions for the first few terms are:

0: ()

1: (1)

3: (2,1)

6: (2,3,1)

11: (3,1,5,2)

17: (4,2,3,7,1)

Representatives of corresponding Golomb rulers are:

{0}

{0,1}

{0,2,3}

{0,2,5,6}

{0,3,4,9,11}

{0,4,6,9,16,17}

(End)

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

a(n-2), for n >= 3, is the number of independent entries of a bisymmetric n X n matrix B_n with B_n[1,1] and B_n[n,n] fixed. Hence a(n-2) = A002620(n+1) - 2. See the Jul 07 2015 comment on A002620. For n=1 and n=2 this matrix B_n is fixed. Bisymmetric matrices B_n, with B_n[1,1] and B_n[n,n] fixed, are, for n >= 3, determined by giving the a(n-2) entries for [1,2], ...., [1,n-1]; [2,2], ..., [2,n-1]; [3,3], ..., [3,n-2]; ..., [ceiling(n/2),n-(ceiling(n/2)-1)]. - Wolfdieter Lang, Aug 16 2015

a(n-1) is the largest possible n-th element in an additive basis of order 2. - Charles R Greathouse IV, May 05 2020

A graph with e edges is "graceful" if its nodes can be labeled with distinct integers in {0,1,...,e} so that, if each edge is labeled with the absolute difference between the labels of its endpoints, then the e edges have the distinct labels 1, 2, ..., e.

Equivalently, maximum m for which there's a restricted difference basis with respect to m with n elements. A "difference basis w.r.t. m" is a set of integers such that every integer from 1 to m is a difference between two elements of the set. A "restricted" difference basis is one in which the smallest element is 0 and the largest is m.

a(n) is also the length of an optimal ruler with n marks. For definitions see A103294. For example, a(6)=13 is the length of the optimal rulers with 6 marks, {[0, 1, 6, 9, 11, 13], [0, 2, 4, 7, 12, 13], [0, 1, 4, 5, 11, 13], [0, 2, 8, 9, 12, 13], [0, 1, 2, 6, 10, 13], [0, 3, 7, 11, 12, 13]}. Also n = 1 + A103298(a(n)). - Peter Luschny, Feb 28 2005

If the conjecture is true that an optimal ruler with more than 12 segments is a Wichmann ruler then the sequence continues 232, 251, 270, 289, 308, 327, ... - Peter Luschny, Oct 09 2011 [updated to take the verifications of Robison into account, Oct 01 2015]

A permutation of the natural numbers.

Except for initial terms, rows 1 to 4 are A002061, A002522, A014206, A059100 and columns 1 to 4 are A002620, A024206, A014616, A004116.

This is the interspersion of the fractal sequence A167430; i.e., row n of this array consists of the numbers k such that n=A167430(k). - Clark Kimberling, Nov 03 2009

Conjectures: a(2^n)=a(2^(n+1)+1)=A033638(n); a(2^n-1)=a(3*2^n-1)=1.

The Gi1 and Gi2 triangle sums, see A180662 for their definitions, of Sierpinski's triangle A047999 equal this sequence. The Gi1 and Gi2 sums can also be interpreted as (i + 4*j = n) and (4*i + j = n) sums, see the Northshield reference. Some A112970(2^n-p) sequences, 0<=p<=32, lead to known sequences, see the crossrefs. - Johannes W. Meijer, Jun 05 2011

a(n) is not A173642, a compact Bohr-Stoner model (1924), modified by Charles Janet in 1930. The good distribution is A168208.

Only sequences N16(n) in A234398 are used:

N16(1)= 1 followed by 2's = A040000,

N16(2)= 1, 2, 3, 4, 5, followed by 6's = A101272,

N16(3)= 1 to 9, followed by 10's,

N16(4)= 1 to 13, followed by 14's, etc.

The distribution by rows are in the example.

The N16(n)'s are respectively on columns (hence triangle T)

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, A002620(n+2)

3, 5, 8, 11, 15, 19, 24, 29, 35, A024206(n+2)

7, 10, 14, 18, 23, 28, 34, A014616(n+3)

13, 17, 22, 27, 33, A004116(n+4)

21, 26, 32,

31, etc.

See A163255.

Antidiagonals give the natural numbers A000027, like rows sums in the example.

A033638=1, 1, 2, 3, 5, 7,... is upon the triangle T.

The natural numbers, based on quarter-squares (A002620 and A033638); every natural number occurs exactly twice.