A004482 -id:A004482 - cp's OEIS frontend

A004481 Table of Sprague-Grundy values for Wythoff's game (Wyt Queens) read by antidiagonals.

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

Offset: 0

N. J. A. Sloane

T(a,b) = T(b,a).

			Table begins
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   1,  2,  0,  4,  5,  3,  7,  8,  6, 10, 11,  9, ...
   2,  0,  1,  5,  3,  4,  8,  6,  7, 11,  9, ...
   3,  4,  5,  6,  2,  0,  1,  9, 10, 12, ...
   4,  5,  3,  2,  7,  6,  9,  0,  1, ...
   5,  3,  4,  0,  6,  8, 10,  1, ...
   6,  7,  8,  1,  9, 10,  3, ...
   7,  8,  6,  9,  0,  1, ...
   8,  6,  7, 10,  1, ...
   9, 10, 11, 12, ...
  10, 11,  9, ...
  11,  9, ...
  12, ...
  ...

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.
Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, date?
R. K. Guy, The unity of combinatorics, Proc. 25th Iranian Math. Conf, Tehran, (1994), Math. Appl 329 129-159, Kluwer Dordrecht 1995, Math. Rev. 96k:05001.

Vincenzo Librandi, Table of n, a(n) for n = 0..5049
Uri Blass and Aviezri S. Fraenkel, The Sprague-Grundy function for Wythoff's game, Theoretical Computer Science 75.3 (1990): 311-333. See Table 2.
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.
A. Dress, A. Flammenkamp and N. 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.
Rémy Sigrist, Colored representation of T(x,y) for x = 0..999 and y = 0..999 (where the hue is function of T(x,y) and black pixels correspond to zeros)
Rémy Sigrist, PARI program for A004481

A004482-A004487 are rows 1 to 6. Cf. A047708 (main diagonal).
See A317205 for triangle of values on or below main diagonal.
Similar to but different from A004489.
T(a, b)=0 iff A018219(a, b)=0 iff A002251(a)=b.

mex[list_] := mex[list] = Min[Complement[Range[0, Length[list]], list]];
move[Wnim, {a_, b_}] := move[Wnim, {a, b}] =
   Union[Table[{i, b}, {i, 0, a - 1}], Table[{a, i}, {i, 0, b - 1}],
    Table[{a - i, b - i}, {i, 1, Min[a, b]}]];
SpragueGrundy[game_, list_] := SpragueGrundy[game, list] =
   mex[SpragueGrundy[game, #] & /@ move[game, list]];
t[n_, m_] := SpragueGrundy[Wnim, {n - 1, m - 1}];
Flatten@Table[t[n - m + 1, m], {n, 11}, {m, n}] (* Birkas Gyorgy, Apr 19 2011 *)

PARI
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See Links section.
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A274315 First row of infinite Sudoku-type array A269526.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 11, 13, 8, 14, 18, 7, 20, 19, 9, 12, 24, 26, 23, 25, 29, 16, 15, 35, 31, 38, 40, 37, 39, 41, 17, 43, 42, 47, 46, 45, 52, 27, 21, 22, 51, 58, 53, 60, 50, 56, 62, 64, 63, 67, 66, 68, 73, 72, 59, 74, 28, 77, 76, 70, 71, 30, 87, 32, 83, 84, 33, 34, 89, 88, 92, 91, 36, 98, 93, 96

Offset: 1

N. J. A. Sloane, Jun 29 2016

nonn

Conjectured to be a permutation of the natural numbers.
It would be nice to have a formula or recurrence.  Note that the first row of the analogous array corresponding to the Wythoff game, A004482, does have a simple formula.
See A295563 for much more about this sequence. - N. J. A. Sloane, Mar 10 2019

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A004482, A269526, A274316, A274317, A274318, A295563.

A296339 On an infinite 60-degree sector of hexagonal graph paper, fill in cells by antidiagonals so that each contains the least nonnegative integer such that no line of edge-adjacent cells contains a repeated term.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 10 2017

To find the number to enter in a cell (assuming the sector is oriented as in the illustration in the link), look at all the numbers in the cells directly above the cell, in the cells to the "North-West", and in the cells to the "South-West", and take their "mex" (the smallest missing number).

The 0-cells in the array all lie on a perfectly straight line (in contrast to the situation in A274528). Also a(n) = 0 iff n = 2*m*(m+1) for some m.

			The initial rows are as follows (however, this does not show the adjancies between the cells correctly - for that, see the illustration in the link):
   0;
   1,  2;
   2,  0,  1;
   3,  1,  2,  4;
   4,  5,  0,  3,  6;
   5,  3,  4,  6,  7,  8;
   6,  4,  5,  0,  3,  9,  7;
   7,  8,  3,  1,  2,  4,  5,  9;
   8,  6,  7,  2,  0,  1,  9,  4,  3;
   9,  7,  8,  5,  1,  2,  6, 10, 11, 12;
  10, 11,  6,  9,  4,  0, ...
  ...
For example, referring to the illustration in the link and NOT to the triangle here, consider the first 5 in the array. The reason this is 5 is because in the column of cells above that cell we can see 2,0,1, to the NW we see 3, and to the SW we see 4, and the smallest missing number is 5.

Rémy Sigrist, Rows n = 0..200, flattened
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, Colored representation of the first 2^9 rows of the triangle
Rémy Sigrist, PARI program for A296339
N. J. A. Sloane, Illustration of initial rows of the sector.

Two analogs of this for an infinite square chessboard are A269526 (which uses positive numbers) and A274528 (which uses nonnegative numbers).
For the right edge see A296340.
The second column is A004483. - Rémy Sigrist, Dec 11 2017
The third and fourth columns are A004482 and A298801.
See also A274820.

ab = Table[0, {13}];
nw = ab;
A296339 = Reap[For[s = 1, s <= Length[ab], s++, sw = 0; For[c = 1, c <= s, c++, x = BitOr[ab[[c]], BitOr[nw[[s-c+1]], sw]]; v = IntegerExponent[x+1, 2]; Sow[v]; p = 2^v; sw += p; ab[[c]] += p; nw[[s-c+1]] += p]]][[2, 1]] (* Jean-François Alcover, Dec 18 2017, after Rémy Sigrist *) (* I changed the first line, which was ab = Table[0, 13];, to make this compatible with older versions of MMA - N. J. A. Sloane, Feb 03 2018 *)

PARI
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See Links section.
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More terms from Rémy Sigrist, Dec 11 2017

A295563 First row of array in A274528.

Original entry on oeis.org

0, 2, 1, 5, 3, 4, 9, 10, 12, 7, 13, 17, 6, 19, 18, 8, 11, 23, 25, 22, 24, 28, 15, 14, 34, 30, 37, 39, 36, 38, 40, 16, 42, 41, 46, 45, 44, 51, 26, 20, 21, 50, 57, 52, 59, 49, 55, 61, 63, 62, 66, 65, 67, 72, 71, 58, 73, 27, 76, 75, 69, 70, 29, 86, 31, 82, 83, 32, 33, 88, 87, 91, 90, 35, 97, 92, 95

Offset: 0

N. J. A. Sloane, Nov 29 2017

nonn

Note this is not a list, this gives the Grundy-value at (0,n) of the Lonely Queens array A274528 regarded as a game.
It would be nice to have a formula or recurrence.  Note that the first row of the analogous array corresponding to the Wythoff game, A004482, does have a simple formula.
The points seem to fall on or close to two lines, of slopes about 0.48 and 1.29 (what are these slopes?). See A295564, A295565, A295566, A295567 for the X- and Y-coordinates of the points on the two lines.

Rémy Sigrist, Table of n, a(n) for n = 0..99999 (terms 0..1999 from N. J. A. Sloane)
Rémy Sigrist, PARI program for A295563

Cf. A004482, A269526, A274528, A295564, A295565, A295566, A295567.

Equals A274315 - 1.

PARI
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See Links section.
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A330396 Permutation of the nonnegative integers partitioned into triples [3k+2, 3k+1, 3*k] for k >= 0.

Original entry on oeis.org

2, 1, 0, 5, 4, 3, 8, 7, 6, 11, 10, 9, 14, 13, 12, 17, 16, 15, 20, 19, 18, 23, 22, 21, 26, 25, 24, 29, 28, 27, 32, 31, 30, 35, 34, 33, 38, 37, 36, 41, 40, 39, 44, 43, 42, 47, 46, 45, 50, 49, 48, 53, 52, 51, 56, 55, 54, 59, 58, 57, 62, 61, 60, 65, 64, 63, 68, 67, 66, 71, 70, 69, 74, 73, 72, 77, 76, 75, 80, 79, 78, 83, 82

Offset: 0

Guenther Schrack, Mar 03 2020

nonn

Partition the nonnegative integer sequence into triples starting with (0,1,2); transpose the first and third elements of the triple, repeat for all triples.
A self-inverse sequence: a(a(n)) = n.
The sequence is an interleaving of A016789 with A016777 and with A008585, in that order.

Guenther Schrack, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the nonnegative integers
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A001477, A064429, A236348, A004482, A004483.
Fixed point sequence: A016777.
Relationships:
  a(n) = a(n-1) - 1 + 6*A079978(n).
  a(n) = 2*a(n-1) - a(n-2) + 6*A049347(n).
  a(n) = A074066(n+2) - 2.
  a(n) = A113655(n+1) - 1.

a = zeros(1,10000);
w = (-1+sqrt(-3))/2;
fprintf('0 2\n');
for n = 1:10000
   a(n) = int64((3*n + 2*w^(2*n)*(w + 2) + 2*w^n*(1 - w))/3);
   fprintf('%i %i\n',n,a(n));
end

G.f.: (2 - x - x^2 + 3*x^3)/((x-1)^2*(1 + x + x^2)). [corrected by Georg Fischer, Apr 17 2020]
Linear recurrence: a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
Simple recursion: a(n) = a(n-3) + 3 for n > 2 with a(0) = 2, a(1) = 1, a(2) = 0.
Negative domain: a(-n) = -(a(n-1) + 1).
Explicit formulas:
  a(n) = n + 2 - 2*(n mod 3).
  a(n) = 2 - n + 6*floor(n/3).
  a(n) = n + 2*(w^(2*n)*(2 + w) + w^n*(1 - w))/3 where w = (-1 + sqrt(-3))/2.

A059249 Tersum n + (n-1); write n and n-1 in base 3 and add mod 3 with no carries.

Original entry on oeis.org

1, 0, 5, 7, 6, 2, 4, 3, 17, 19, 18, 23, 25, 24, 20, 22, 21, 8, 10, 9, 14, 16, 15, 11, 13, 12, 53, 55, 54, 59, 61, 60, 56, 58, 57, 71, 73, 72, 77, 79, 78, 74, 76, 75, 62, 64, 63, 68, 70, 69, 65, 67, 66, 26, 28, 27, 32, 34, 33, 29, 31, 30, 44, 46, 45, 50, 52, 51, 47, 49, 48, 35

Offset: 1

Henry Bottomley, Jan 22 2001

			a(21)=14 since 21 and 20 are written in base 3 as 210 and 202 and so their tersum is 112 in base 3, i.e. 9+3+2=14.

Alois P. Heinz, Table of n, a(n) for n = 1..19682

Cf. A004482, A004488, A004489, A038712.

A071770 Tersum n + [n/3] (answer recorded in base 10).

Original entry on oeis.org

0, 1, 2, 4, 5, 3, 8, 6, 7, 12, 13, 14, 16, 17, 15, 11, 9, 10, 24, 25, 26, 19, 20, 18, 23, 21, 22, 36, 37, 38, 40, 41, 39, 44, 42, 43, 48, 49, 50, 52, 53, 51, 47, 45, 46, 33, 34, 35, 28, 29, 27, 32, 30, 31, 72, 73, 74, 76, 77, 75, 80, 78, 79, 57, 58, 59, 61, 62, 60, 56, 54, 55, 69, 70

Offset: 0

Paul D. Hanna, Jun 04 2002

Alois P. Heinz, Table of n, a(n) for n = 0..19682
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
Index entries for sequences that are permutations of the nonnegative integers

Cf. A004482.

Inverse permutation to A370932.

a:= n-> (l-> add(irem(l[i]+l[i-1], 3)*3^(i-2),
         i=2..nops(l)))([convert(n, base, 3)[], 0]):
seq(a(n), n=0..73);  # Alois P. Heinz, Aug 07 2024

a(n)={if(n==0, 0, fromdigits((digits(n,3) + concat([0],digits(n\3,3)))%3,3))} \\ Andrew Howroyd, Aug 06 2024

from sympy.ntheory import digits
def a(n):
    d = digits(n, 3)[1:]
    return int(str(d[0]) + "".join(str((d[i]+d[i-1])%3) for i in range(1, len(d))), 3)
print([a(n) for n in range(75)]) # Michael S. Branicky, Aug 07 2024

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. See A004482 for references.

a(27) corrected by Sean A. Irvine, Aug 06 2024

A004498 Tersum n + 9.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Cf. A004482, A004483, A004492, A004493, A004494, A004495, A004496, A004497.

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.

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)

A004499 Tersum n + 10.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Cf. A004482, A004483, A004492, A004493, A004494.

Cf. A004495, A004496, A004497, A004498.

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.

cp's OEIS Frontend

A004481 Table of Sprague-Grundy values for Wythoff's game (Wyt Queens) read by antidiagonals.

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A274315 First row of infinite Sudoku-type array A269526.

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A296339 On an infinite 60-degree sector of hexagonal graph paper, fill in cells by antidiagonals so that each contains the least nonnegative integer such that no line of edge-adjacent cells contains a repeated term.

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A295563 First row of array in A274528.

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A330396 Permutation of the nonnegative integers partitioned into triples [3*k+2, 3*k+1, 3*k] for k >= 0.

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MATLAB

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A059249 Tersum n + (n-1); write n and n-1 in base 3 and add mod 3 with no carries.

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A071770 Tersum n + [n/3] (answer recorded in base 10).

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Maple

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Python

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A004498 Tersum n + 9.

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A298801 Fourth column of triangular array in A296339.

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A330396 Permutation of the nonnegative integers partitioned into triples [3k+2, 3k+1, 3*k] for k >= 0.