A003057 -id:A003057 - cp's OEIS frontend

A158440 Triangle T(n,k) read by rows: row n contains n times n+1 followed by n 1's.

2, 1, 3, 3, 1, 1, 4, 4, 4, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 1, 6, 6, 6, 6, 6, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, 8, 8, 8, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 9, 9, 9, 9, 1, 1, 1, 1, 1, 1, 1, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Paul Curtz, Mar 19 2009

These are essentially the unreduced numerators of the coefficients of P(2n-1,x) of A130679, with denominators represented by A122197.

			The triangle has 2n columns in row n. It starts:
  2, 1;
  3, 3, 1, 1;
  4, 4, 4, 1, 1, 1;
  5, 5, 5, 5, 1, 1, 1, 1;

Cf. A003057.

Flatten[Table[Join[PadRight[{},n,n+1],PadRight[{},n,1]],{n,12}]] (* Harvey P. Dale, Feb 18 2013 *)

Edited by R. J. Mathar, Apr 09 2009

A225203 Table T(n,k) composed of rows equal to: n * (the characteristic function of the multiples of (n+1)), read by downwards antidiagonals.

Original entry on oeis.org

1, 0, 2, 1, 0, 3, 0, 0, 0, 4, 1, 2, 0, 0, 5, 0, 0, 0, 0, 0, 6, 1, 0, 3, 0, 0, 0, 7, 0, 2, 0, 0, 0, 0, 0, 8, 1, 0, 0, 4, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 2, 3, 0, 5, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 13

Offset: 1

Richard R. Forberg, May 01 2013

Column k =1 of the table is the integers, from n=1 in row 1.
The n-th row of the table is a repeating pattern, starting with the value of n followed by n instances of zero, as created by the characteristic function of the multiples of (n+1).
Sums of the antidiagonals produce A065608.
Row 1 is A059841, row 2 = 2*A079978, row 3 = 3*A121262, row 4 = 4*A079998, row 5 = 5*A079979, row 6 = 6*A082784, row 7 = 7*|A014025|. - Boris Putievskiy, May 08 2013

			Table begins:
  1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0 ...
  2,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,0 ...
  3,0,0,0,3,0,0,0,3,0,0,0,3,0,0,0,3,0 ...
  4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4,0,0 ...
  5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0 ...
  6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0 ...
  7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0 ...
  8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0 ...
  9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0 ...

Cf. A065608, A002024, A002260, A003057, A059841, A079978, A121262, A079998, A079979, A082784, A014025.

From Boris Putievskiy, May 08 2013: (Start)
As table T(n,k) = n*(floor((n+k)/(n+1))-floor((n+k-1)/(n+1))).
As linear sequence a(n) = A002260(n)*(floor(A003057(n))/(A002260(n)+1)-floor(A002024(n))/(A002260(n)+1)); a(n) = i*(floor((t+2)/(i+1))-floor((t+1)/(i+1))), where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)

More terms from Jason Yuen, Feb 22 2025

A349098 Where ones occur in A349085. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/v + 1/w + 1/x + 1/y + 1/z, 0 < v < w < x < y < z.

Original entry on oeis.org

2211, 3402, 3403, 3912, 3914, 3916, 4656, 5048, 5565, 5867, 6326, 8505, 8507, 8910, 9293, 9294, 9313, 9578, 9586, 9587, 9588, 10585, 11002, 11017, 11308, 11317, 11322, 11324, 11935, 12242, 12244, 12245, 12558, 12560, 13193, 13194, 13692, 13694, 13834

Offset: 1

Jud McCranie, Dec 31 2021

For index k, p/q = A002260(k)/A003057(k).

			11002 is a term because A349085(11002) = 1, indicating that 124/149 = 1/v + 1/w + 1/x + 1/y + 1/z has a unique solution, namely 1/2 + 1/4 + 1/13 + 1/189 + 1/1464372.

Cf. A349085, A002260, A003057, A349090, A349095, A349096, A349097.

A381564 2-tone chromatic number of a path with n-2 vertices joined to two adjacent vertices.

Original entry on oeis.org

8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16

Offset: 4

Allan Bickle, Feb 27 2025

nonn

The 2-tone chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of vertices at distance 2 have two common colors.

The graphs are maximal planar.

			The central vertices each have two disjoint labels.  All vertices on the path require distinct pairs.
The colorings for small paths are shown below.
  12-34
  12-34-15
  12-34-15-23
  12-34-15-23-14
  12-34-15-23-14-25
  12-34-15-23-14-25-13
  12-34-15-23-14-25-13-24
  12-34-15-23-14-25-13-24-35

Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013) 171-190.
Allan Bickle, 2-Tone Coloring of Planar Graphs, Bull. Inst. Combin. Appl. 103 (2025) 114-129.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.

Cf. A003057, A351120 (pair coloring).

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar), A381563 (double wheels).

a(n) = ceiling((9 + sqrt(8*n - 15))/2) for n > 8.

A381565 2-tone chromatic number of a particular class of planar graphs with 3n+3 vertices.

Original entry on oeis.org

5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21

Offset: 1

Allan Bickle, Feb 27 2025

nonn

The 2-tone chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of vertices at distance 2 have two common colors.

The graphs are formed by replacing each edge of K_3 by n disjoint paths with length 2, resulting in 3n+3 vertices. These graphs have large 2-tone chromatic number relative to their maximum degree of 2t.

			For n=1, the graph is a 6-cycle, which has a 2-tone 5-coloring -12-34-15-32-14-35-.  Thus a(1) = 5.

Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013) 171-190.
Allan Bickle, 2-Tone Coloring of Planar Graphs, Bull. Inst. Combin. Appl. 103 (2025) 114-129.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.
D. W. Cranston and H. LaFayette, The t-tone chromatic number of classes of sparse graphs, Australas. J. Combin. 86 (2023) 458-476.

Cf. A003057, A351120 (pair coloring).

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar).

a(n) = ceiling(1.5 + sqrt(6*n + 6.25)) for n < 18.

a(n) = ceiling(0.5 + sqrt(6*n + 24.25)) for n > 6.

cp's OEIS Frontend

A158440 Triangle T(n,k) read by rows: row n contains n times n+1 followed by n 1's.

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A225203 Table T(n,k) composed of rows equal to: n * (the characteristic function of the multiples of (n+1)), read by downwards antidiagonals.

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A349098 Where ones occur in A349085. These correspond to rationals, 0 < p/q < 1, that have a unique solution, p/q = 1/v + 1/w + 1/x + 1/y + 1/z, 0 < v < w < x < y < z.

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A381564 2-tone chromatic number of a path with n-2 vertices joined to two adjacent vertices.

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Keywords

Comments

Examples

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Crossrefs

Formula

A381565 2-tone chromatic number of a particular class of planar graphs with 3n+3 vertices.

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Formula