A296339 -id:A296339 - cp's OEIS frontend

A296340 The right edge of the array in A296339.

0, 2, 1, 4, 6, 8, 7, 9, 3, 12, 14, 16, 5, 17, 11, 18, 10, 24, 23, 25, 27, 26, 30, 29, 13, 32, 34, 31, 35, 38, 15, 39, 40, 44, 20, 45, 37, 49, 48, 51, 52, 53, 55, 19, 22, 21, 61, 59, 60, 62, 56, 65, 68, 58, 69, 71, 70, 72, 74, 28, 79, 77, 81, 78, 83, 80, 86, 88

Offset: 0

N. J. A. Sloane, Dec 10 2017

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..9999
Rémy Sigrist, PARI program for A296340

Cf. A296339.

PARI
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More terms from Rémy Sigrist, Dec 11 2017

A298801 Fourth column of triangular array in A296339.

Original entry on oeis.org

4, 3, 6, 0, 1, 2, 5, 9, 12, 7, 8, 15, 10, 11, 18, 13, 14, 21, 16, 17, 24, 19, 20, 27, 22, 23, 30, 25, 26, 33, 28, 29, 36, 31, 32, 39, 34, 35, 42, 37, 38, 45, 40, 41, 48, 43, 44, 51, 46, 47, 54, 49, 50, 57, 52, 53, 60, 55, 56, 63, 58, 59, 66, 61, 62, 69, 64, 65, 72, 67, 68, 75, 70, 71, 78, 73, 74, 81, 76

Offset: 0

N. J. A. Sloane, Feb 02 2018

nonn

This was the first column of A296339 for which no simple formula was known (cf. A004483, A004482). (Since these are Grundy values for a certain game, there is a complicated recurrence involving the whole triangle.) The formula below matches the data, and is fairly short (but ugly).

Cf. A296339, A004482, A004483, A296340.

It appears that for n >= 8, a(n) = tersum(n,1) + 6 if n == 2 (mod 3), otherwise tersum(n,1) - 3.
Conjectures from Colin Barker, Feb 03 2018: (Start)
G.f.: (4 - x + 3*x^2 - 10*x^3 + 2*x^4 - 2*x^5 + 9*x^6 + 3*x^7 + 2*x^8 - 8*x^9 - 3*x^10 + 4*x^11) / ((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>11.
(End)

A274820 Spiral constructed on the nodes of the infinite triangular net in which each term is the least nonnegative integer such that no diagonal contains a repeated term.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 0, 3, 0, 4, 3, 5, 3, 4, 5, 3, 4, 6, 5, 6, 7, 4, 6, 5, 7, 6, 3, 0, 6, 5, 7, 0, 6, 7, 5, 4, 8, 1, 3, 6, 8, 1, 9, 7, 8, 2, 4, 9, 8, 2, 10, 11, 8, 9, 10, 12, 3, 8, 9, 7, 10, 9, 2, 4, 8, 5, 10, 2, 11, 9, 11, 0, 10, 7, 8, 6, 0, 9, 7, 10, 12, 7, 1, 4, 8, 5, 11, 1, 10, 12, 9, 5, 11, 10, 13, 12, 11, 13, 14

Offset: 0

Omar E. Pol, Jul 09 2016

Also spiral constructed on the infinite hexagonal grid in which each term is the least nonnegative integer such that no diagonal of successive adjacent cells contains a repeated term. Every number is located in the center of a hexagonal cell. Every cell is also the center of three diagonals of successive adjacent cells.

Presumably every line of cells with slope a multiple of 60 degrees (not necessarily passing through the central cell) is a permutation of the nonnegative numbers. See A296343-A296348 for the spokes through the central cell. - N. J. A. Sloane, Dec 12 2017

			Illustration of initial terms as a spiral:
.
.                   9 - 4 - 2 - 8 - 7
.                  /                 \
.                 8   3 - 6 - 7 - 5   9
.                /   /             \   \
.               2   0   5 - 3 - 4   6   1
.              /   /   /         \   \   \
.            10   6   3   1 - 2   0   4   8
.            /   /   /   /     \   \   \   \
.          11   5   4   2   0 - 1   3   7   6
.            \   \   \   \         /   /   /
.             8   7   5   1 - 2 - 0   6   3
.              \   \   \             /   /
.               9   0   3 - 4 - 6 - 5   1
.                \   \                 /
.                10   6 - 7 - 5 - 4 - 8
.                  \
.                  12 - 3 - 8 - 9 - 7
.

Rémy Sigrist, Table of n, a(n) for n = 0..120400
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Rémy Sigrist, PARI program for A274820
Rémy Sigrist, Colored illustration of the first 200 windings of the spiral (where the color is a function of a(n))
N. J. A. Sloane, Illustration of initial terms drawn as a spiral on the hexagonal grid (the starting cell is marked in black).

Cf. A001477, A269526, A274528 (square array), A274641 (spiral on the square grid), A274650 (right triangle), A274821, A274920, A274921, A275606, A275610, A296339.
A296342 says when n first appears.
See A296343-A296348 for the spokes.

PARI
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a(n) = A274821(n) - 1.

A004482 Tersum n + 1 (answer recorded in base 10).

Original entry on oeis.org

1, 2, 0, 4, 5, 3, 7, 8, 6, 10, 11, 9, 13, 14, 12, 16, 17, 15, 19, 20, 18, 22, 23, 21, 25, 26, 24, 28, 29, 27, 31, 32, 30, 34, 35, 33, 37, 38, 36, 40, 41, 39, 43, 44, 42, 46, 47, 45, 49, 50, 48, 52, 53, 51, 55, 56, 54, 58, 59, 57, 61, 62

Offset: 0

N. J. A. Sloane

Tersum m + n: write m and n in base 3 and add mod 3 with no carries; e.g., 5 + 8 = "21" + "22" = "10" = 1.

Sprague-Grundy values for game of Wyt Queens.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), Article P1.52.
Andreas Dress, Achim Flammenkamp, and Norbert Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).
Gabriel Nivasch, More on the Sprague-Grundy function for Wythoff's game, pages 377-410 in "Games of No Chance 3", MSRI Publications Volume 56, 2009. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

This sequence is row 1 of table A004481.
a(n) = A061347(n+1) + n.
Third column of triangle A296339.

LinearRecurrence[{1,0,1,-1},{1,2,0,4},70] (* or *) Table[3*Floor[n/3]+ Mod[ n+1,3],{n,0,70}] (* Harvey P. Dale, Nov 29 2014 *)

Periodic with period 3 and saltus 3: a(n) = 3*floor(n/3) + ((n+1) mod 3).
a(n) = n - 2*cos(2*(n+1)*Pi/3). - Wesley Ivan Hurt, Sep 29 2017
Sum_{n>=3} (-1)^(n+1)/a(n) = 1/2 - log(2)/3. - Amiram Eldar, Aug 21 2023

More terms from Erich Friedman

A004483 Tersum n + 2.

Original entry on oeis.org

2, 0, 1, 5, 3, 4, 8, 6, 7, 11, 9, 10, 14, 12, 13, 17, 15, 16, 20, 18, 19, 23, 21, 22, 26, 24, 25, 29, 27, 28, 32, 30, 31, 35, 33, 34, 38, 36, 37, 41, 39, 40, 44, 42, 43, 47, 45, 46, 50, 48, 49, 53, 51, 52, 56, 54, 55, 59, 57, 58, 62, 60, 61, 65, 63, 64, 68, 66, 67

Offset: 0

N. J. A. Sloane

Tersum m + n: write m and n in base 3 and add mod 3 with no carries; e.g., 5 + 8 = "21" + "22" = "10" = 1.

Also Sprague-Grundy values for game of Wyt Queens.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

Andreas Dress, Achim Flammenkamp, and Norbert Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

This sequence is row 2 of table A004481.

Second column of triangle in A296339.

a[n_] := If[Divisible[n, 3], n+2, n-1]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Oct 25 2013 *)
LinearRecurrence[{1,0,1,-1},{2,0,1,5},70] (* Harvey P. Dale, Feb 07 2018 *)

Periodic with period and saltus 3: a(n) = 3*floor(n/3) + ((n+2) mod 3).
a(n) = n + 2*cos(2*n*Pi/3). - Wesley Ivan Hurt, Sep 27 2017
From R. J. Mathar, Dec 14 2017: (Start)
G.f.: ( 2+x^2+2*x^3-2*x ) / ( (1+x+x^2)*(x-1)^2 ).
a(n) = n + A099837(n) if n > 0. (End)
Sum_{n>=2} (-1)^n/a(n) = 2*Pi/(3*sqrt(3)) + log(2)/3 - 1/2. - Amiram Eldar, Aug 21 2023

Edited by N. J. A. Sloane at the suggestion of Philippe Deléham, Nov 20 2007

A362312 Sierpinski triangle read by rows and filled in the greedy way such that each row, each diagonal and each antidiagonal contains distinct nonnegative values.

Original entry on oeis.org

0, 1, 2, 2, 1, 3, 0, 2, 4, 4, 3, 5, 1, 0, 6, 6, 0, 1, 5, 7, 3, 4, 2, 5, 8, 6, 9, 8, 7, 9, 4, 3, 8, 10, 3, 4, 11, 11, 5, 6, 0, 1, 3, 4, 10, 12, 2, 5, 13, 13, 6, 4, 1, 0, 7, 5, 12, 14, 5, 6, 7, 2, 3, 9, 15, 15, 7, 8, 1, 9, 0, 2, 5, 6, 4, 10, 11, 12, 13, 14, 16, 16, 14

Offset: 0

Rémy Sigrist, Apr 15 2023

This sequence is a variant of A296339.

The n-th row has A001316(n) terms, the first one being n and the last one being A361740(n).

			Sierpinski triangle begins (with dots denoting empty places):
                                  0
                                1   2
                              2   .   1
                            3   0   2   4
                          4   .   .   .   3
                        5   1   .   .   0   6
                      6   .   0   .   1   .   5
                    7   3   4   2   5   8   6   9
                  8   .   .   .   .   .   .   .   7
                9   4   .   .   .   .   .   .   3   8
             10   .   3   .   .   .   .   .   4   .  11
           11   5   6   0   .   .   .   .   1   3   4  10
         12   .   .   .   2   .   .   .   5   .   .   .  13
       13   6   .   .   4   1   .   .   0   7   .   .   5  12
     14   .   5   .   6   .   7   .   2   .   3   .   9   .  15
   15   7   8   1   9   0   2   5   6   4  10  11  12  13  14  16

Rémy Sigrist, Table of n, a(n) for n = 0..6560 (rows for n = 0..255 flattened)
Rémy Sigrist, Colored representation of the first 512 rows (the hue is function of the terms, black pixels denote 0's, white pixels denote empty places)
Rémy Sigrist, C++ program

Cf. A001316, A296339, A361740 (right border), A362313 (least values).

A335490 Isosceles triangle read by rows in which each term is the least positive integer satisfying the condition that no row, diagonal, or antidiagonal contains a repeated term.

Original entry on oeis.org

1, 2, 3, 3, 1, 2, 4, 2, 3, 5, 5, 6, 1, 4, 7, 6, 4, 5, 7, 8, 9, 7, 5, 6, 1, 4, 10, 8, 8, 9, 4, 2, 3, 5, 6, 10, 9, 7, 8, 3, 1, 2, 10, 5, 4, 10, 8, 9, 6, 2, 3, 7, 11, 12, 13, 11, 12, 7, 10, 5, 1, 9, 8, 6, 14, 15, 12, 10, 11, 13, 6, 4, 14, 7, 9, 8, 16, 17, 13, 11

Offset: 1

Alec Jones and Peter Kagey, Sep 12 2020

The n-th instance of 1 occurs at index A001844(n-1).

Records occur at 1, 2, 3, 7, 10, 12, 15, 20, 21, 27, 53, 54, 55, 65, ...

			Triangle begins:
       1
      2 3
     3 1 2
    4 2 3 5
   5 6 1 4 7
  6 4 X ...
The value for X is 5 because 1, 2, and 3 are on the diagonal; 4 and 6 are on the antidiagonal; and 4 and 6 are in the row. Therefore 5 is the smallest value that can be inserted so that no diagonal, antidiagonal, or row contains a repeated term.

Analogs for other tilings: A269526 (square), A334049 (triangular).

Cf. A001844, A274650, A274651, A288530, A288531, A296339.

a(n) = A296339(n-1) + 1. - Rémy Sigrist, Sep 13 2020

cp's OEIS Frontend

A296340 The right edge of the array in A296339.

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A298801 Fourth column of triangular array in A296339.

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A274820 Spiral constructed on the nodes of the infinite triangular net in which each term is the least nonnegative integer such that no diagonal contains a repeated term.

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A004482 Tersum n + 1 (answer recorded in base 10).

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A004483 Tersum n + 2.

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A362312 Sierpinski triangle read by rows and filled in the greedy way such that each row, each diagonal and each antidiagonal contains distinct nonnegative values.

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A335490 Isosceles triangle read by rows in which each term is the least positive integer satisfying the condition that no row, diagonal, or antidiagonal contains a repeated term.

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Examples

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Formula