A013574 -id:A013574 - cp's OEIS frontend

A234943 Array read by antidiagonals: T(i,j) = size of optimal difference triangle set M(i,j).

1, 2, 3, 3, 7, 3, 4, 10, 13, 11, 5, 12, 19, 22, 17, 6, 15, 24, 32, 34, 25, 7, 19, 30, 41, 49, 51, 34, 8, 22, 36, 51, 64, 72, 70, 44, 9, 24, 42, 60, 79, 94, 100, 94, 55, 10, 27, 48, 71

Offset: 1

N. J. A. Sloane, Jan 08 2014

An (n,k) difference triangle set is a set of n blocks of k integers such that the difference sets of the blocks are all disjoint. The "scope" of such a set is defined to be the maximal element, if all blocks are translated such that their least elements are all 0. T(n,k) lists the minimal scope for which an (n,k) difference triangle set exists. - Charlie Neder, Jun 14 2019

			Array begins:
j\i|   1   2   3  4  5  6  7  8  9  10  11  12  13  14  15
---+-----------------------------------------------------------------
1  |   1   2   3  4  5  6  7  8  9  10  11  12  13  14  15  (A000027)
2  |   3   7  10 12 15 19 22 24 27  31  34  36  39  43  46  (A013574)
3  |   6  13  19 24 30 36 42 48 54  60  66  72  78  84  90  (A013575)
4  |  11  22  32 41 51 60 71 80 91 100 111 120 131 140 151  (A013576)
5  |  17  34  49 64 79  (A013577)
6  |  25  51  72 94
7  |  34  70 100
8  |  44  94
9  |  55 121
10 |  72
11 |  85
12 | 106
13 | 127
14 | 151
15 | 177
  A003022 A010896 A010898
       A010895 A010897 A010899

CRC Handbook of Combinatorial Designs, 1996, p. 315. (But beware errors!)

Yeow Meng Chee and Charles J. Colbourn, Constructions for Difference Triangle Sets, arXiv:0712.2553 [cs.IT], 2007.
J. B. Shearer, Difference Triangle Sets: Known optimal solutions.
J. B. Shearer, Difference Triangle Sets: Discoverers.

Rows and columns give A000027, A013574, A013575, A013576, A013577; A003022, A010895, A010896, A010897, A010898, A010899.

For the number of different optimal triangle difference sets see the corresponding array in A234947.

A319279 Numbers that are congruent to {0, 3, 7, 10} mod 12.

Original entry on oeis.org

0, 3, 7, 10, 12, 15, 19, 22, 24, 27, 31, 34, 36, 39, 43, 46, 48, 51, 55, 58, 60, 63, 67, 70, 72, 75, 79, 82, 84, 87, 91, 94, 96, 99, 103, 106, 108, 111, 115, 118, 120, 123, 127, 130, 132, 135, 139, 142, 144, 147, 151, 154, 156, 159, 163, 166, 168, 171, 175, 178

Offset: 1

Jianing Song, Sep 16 2018

Key-numbers of the pitches of a minor seventh chord on a standard chromatic keyboard, with root = 0.

Apart from the offset the same as A013574. - R. J. Mathar, Sep 27 2018

Jianing Song, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): A319451
Seventh chords:
Major seventh chord (F,C): A319280
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): this sequence
Half-diminished seventh chord (B): A319452

[n : n in [0..150] | n mod 12 in [0, 3, 7, 10]]

Select[Range[0, 200], MemberQ[{0, 3, 7, 10}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 7, 10, 12}, 100]

x='x+O('x^99); concat(0, Vec(x^2*(3+x+2*x^2)/((x^2+1)*(x-1)^2)))

a(n) = a(n-4) + 12 for n > 4.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x^2*(3 + x + 2*x^2)/((x^2 + 1)*(x - 1)^2).
a(n) = (6*n - 5 + sqrt(2)*cos(Pi*n/2 + Pi/4))/2.
E.g.f.: ((6x - 5)*e^x + sqrt(2)*cos(x + Pi/4) + 4)/2.

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A234943 Array read by antidiagonals: T(i,j) = size of optimal difference triangle set M(i,j).

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A319279 Numbers that are congruent to {0, 3, 7, 10} mod 12.

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