A004489 -id:A004489 - cp's OEIS frontend

A004488 Tersum n + n.

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

Offset: 0

N. J. A. Sloane

Could also be described as "Write n in base 3, then replace each digit with its base-3 negative" as with A048647 for base 4. - Henry Bottomley, Apr 19 2000
a(a(n)) = n, a self-inverse permutation of the nonnegative integers. - Reinhard Zumkeller, Dec 19 2003
First 3^n terms of the sequence form a permutation s(n) of 0..3^n-1, n>=1; the number of inversions of s(n) is A016142(n-1). - Gheorghe Coserea, Apr 23 2018

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 6561 terms from Alois P. Heinz)
Index entries for sequences related to carryless arithmetic
Index entries for sequences that are permutations of the natural numbers

Column k=0 of A253586, A253587.
Column k=3 of A248813.
Row / column 2 of A325820.
Main diagonal of A004489.
Cf. A048647, A055115, A055116, A055120, A059249, A117966, A117967, A117968, A225901, A242399, A244042, A263273, A289813, A289814, A289815, A289816, A289831, A289838, A300222, A321464.

a004488 0 = 0
a004488 n = if d == 0 then 3 * a004488 n' else 3 * a004488 n' + 3 - d
            where (n', d) = divMod n 3
-- Reinhard Zumkeller, Mar 12 2014

a:= proc(n) local t, r, i;
      t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i *irem(2*irem(t, 3, 't'), 3)
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 07 2011

a[n_] := FromDigits[Mod[3-IntegerDigits[n, 3], 3], 3]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 03 2014 *)

a(n) = my(b=3); fromdigits(apply(d->(b-d)%b, digits(n, b)), b);
vector(67, i, a(i-1))  \\ Gheorghe Coserea, Apr 23 2018

from sympy.ntheory.factor_ import digits
def a(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3) # Indranil Ghosh, Jun 06 2017

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.
a(n) = Sum(3-d(i)-3*0^d(i): n=Sum(d(i)*3^d(i): 0<=d(i)<3)). - Reinhard Zumkeller, Dec 19 2003
a(3*n) = 3*a(n), a(3*n+1) = 3*a(n)+2, a(3*n+2) = 3*a(n)+1. - Robert Israel, May 09 2014

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

Original entry on oeis.org

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
```
See Links section.
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A004493 Tersum n + 4.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A004489 (tersum array).

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {4, 5, 3, 7, 8, 6, 1, 2, 0, 13, 14}, 80] (* Jinyuan Wang, Mar 10 2020 *)

my(table=[4,4,1,4,4,1,-5,-5,-8]); a(n) = n + table[n%9+1]; \\ Kevin Ryde, Apr 05 2021

def tersum(a, b):
  c, pow3 = 0, 1
  while a + b > 0:
    a, ra = divmod(a, 3)
    b, rb = divmod(b, 3)
    c, pow3 = c + pow3*((ra+rb)%3), pow3*3
  return c
def a(n): return tersum(n, 4)
print([a(n) for n in range(58)]) # Michael S. Branicky, Apr 05 2021

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.

A004494 Tersum n + 5.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A004489.

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {5, 3, 4, 8, 6, 7,
2, 0, 1, 14}, 80] (* Jinyuan Wang, Mar 10 2020 *)

# T(n, m) in A004489
def a(n): return T(n, 5)
print([a(n) for n in range(68)]) # Michael S. Branicky, May 03 2021

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.

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.

A343836 Array T(n, k), n, k > 0, read by antidiagonals; the balanced ternary representation of T(n, k) is obtained by adding componentwise (i.e., without carries) the digits in the balanced ternary representations of n and of k.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, May 01 2021

This sequence is similar to A003987 and to A004489.
We use the following table to combine individual digits (this is the balanced ternary addition table read mod 3):
         | T 0 1
      ---+-------
       T | 1 T 0
       0 | T 0 1
       1 | 0 1 T

			Array T(n, k) begins:
  n\k|   0   1   2   3   4    5    6    7    8    9   10   11   12   13
  ---+-----------------------------------------------------------------
    0|   0   1   2   3   4    5    6    7    8    9   10   11   12   13
    1|   1  -1   3   4   2    6    7    5    9   10    8   12   13   11
    2|   2   3  -2  -4  -3   10    8    9   13   11   12    7    5    6
    3|   3   4  -4  -3  -2    8    9   10   11   12   13    5    6    7
    4|   4   2  -3  -2  -4    9   10    8   12   13   11    6    7    5
    5|   5   6  10   8   9   -5   -7   -6  -11  -13  -12   -8  -10   -9
    6|   6   7   8   9  10   -7   -6   -5  -13  -12  -11  -10   -9   -8
    7|   7   5   9  10   8   -6   -5   -7  -12  -11  -13   -9   -8  -10
    8|   8   9  13  11  12  -11  -13  -12   -8  -10   -9   -5   -7   -6
    9|   9  10  11  12  13  -13  -12  -11  -10   -9   -8   -7   -6   -5
   10|  10   8  12  13  11  -12  -11  -13   -9   -8  -10   -6   -5   -7
   11|  11  12   7   5   6   -8  -10   -9   -5   -7   -6  -11  -13  -12
   12|  12  13   5   6   7  -10   -9   -8   -7   -6   -5  -13  -12  -11
   13|  13  11   6   7   5   -9   -8  -10   -6   -5   -7  -12  -11  -13
Array T(n, k) begins in balanced ternary:
  n\k|    0    1   1T   10   11  1TT  1T0  1T1  10T  100  101  11T  110  111
  ---+----------------------------------------------------------------------
    0|    0    1   1T   10   11  1TT  1T0  1T1  10T  100  101  11T  110  111
    1|    1    T   10   11   1T  1T0  1T1  1TT  100  101  10T  110  111  11T
   1T|   1T   10   T1   TT   T0  101  10T  100  111  11T  110  1T1  1TT  1T0
   10|   10   11   TT   T0   T1  10T  100  101  11T  110  111  1TT  1T0  1T1
   11|   11   1T   T0   T1   TT  100  101  10T  110  111  11T  1T0  1T1  1TT
  1TT|  1TT  1T0  101  10T  100  T11  T1T  T10  TT1  TTT  TT0  T01  T0T  T00
  1T0|  1T0  1T1  10T  100  101  T1T  T10  T11  TTT  TT0  TT1  T0T  T00  T01
  1T1|  1T1  1TT  100  101  10T  T10  T11  T1T  TT0  TT1  TTT  T00  T01  T0T
  10T|  10T  100  111  11T  110  TT1  TTT  TT0  T01  T0T  T00  T11  T1T  T10
  100|  100  101  11T  110  111  TTT  TT0  TT1  T0T  T00  T01  T1T  T10  T11
  101|  101  10T  110  111  11T  TT0  TT1  TTT  T00  T01  T0T  T10  T11  T1T
  11T|  11T  110  1T1  1TT  1T0  T01  T0T  T00  T11  T1T  T10  TT1  TTT  TT0
  110|  110  111  1TT  1T0  1T1  T0T  T00  T01  T1T  T10  T11  TTT  TT0  TT1
  111|  111  11T  1T0  1T1  1TT  T00  T01  T0T  T10  T11  T1T  TT0  TT1  TTT

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 1094 (blue denotes negative values, red denotes positive values, dark colors correspond to small values in absolute value)
Wikipedia, Balanced ternary: Addition, subtraction and multiplication and division

Cf. A003987, A004489, A343316.

T(n,k,c=v->centerlift(Mod(v,3))) = { if (n==0 && k==0, return (0), my (d=c(n), t=c(k)); c(d+t)+3*T((n-d)/3, (k-t)/3)) }

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = n.
T(n, n) = -n.

A004500 Tersum n + 11.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,0,0,0,-1,1,0,1,-1,0,0,0,0,-1,1,0,1,-1).

Cf. A004489.

def a(n):
    k, pow3, m = 0, 1, 11
    while n + m > 0:
        n, rn = divmod(n, 3)
        m, rm = divmod(m, 3)
        k, pow3 = k + pow3*((rn+rm)%3), pow3*3
    return k
print([a(n) for n in range(58)]) # Michael S. Branicky, Nov 09 2021

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.
a(n) = A004489(n, 11). - Michel Marcus, Nov 09 2021
G.f.: (20*x^21+x^20-2*x^19-16*x^18-7*x^12+x^11-2*x^10+11*x^9-7*x^3+x^2-2*x+11) / ((x^2+x+1)*(x^18+x^9+1)*(x-1)^2). - Alois P. Heinz, Nov 09 2021

A004501 Tersum n + 12.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 9, 10, 11, 21, 22, 23, 24, 25, 26, 18, 19, 20, 3, 4, 5, 6, 7, 8, 0, 1, 2, 39, 40, 41, 42, 43, 44, 36, 37, 38, 48, 49, 50, 51, 52, 53, 45, 46, 47, 30, 31, 32, 33, 34, 35, 27, 28, 29, 66, 67, 68, 69

Offset: 0

N. J. A. Sloane

nonn

Cf. A004489.

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.

A004502 Tersum n + 13.

Original entry on oeis.org

13, 14, 12, 16, 17, 15, 10, 11, 9, 22, 23, 21, 25, 26, 24, 19, 20, 18, 4, 5, 3, 7, 8, 6, 1, 2, 0, 40, 41, 39, 43, 44, 42, 37, 38, 36, 49, 50, 48, 52, 53, 51, 46, 47, 45, 31, 32, 30, 34, 35, 33, 28, 29, 27, 67, 68, 66, 70

Offset: 0

N. J. A. Sloane

nonn

Cf. A004489.

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

A004488 Tersum n + n.

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

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A004493 Tersum n + 4.

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A004494 Tersum n + 5.

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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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A343836 Array T(n, k), n, k > 0, read by antidiagonals; the balanced ternary representation of T(n, k) is obtained by adding componentwise (i.e., without carries) the digits in the balanced ternary representations of n and of k.

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A004500 Tersum n + 11.

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A004501 Tersum n + 12.

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