A004488 -id:A004488 - cp's OEIS frontend

A248813 A(n,k) is the base-k complement of n; square array A(n,k), n>=0, k>=2, read by antidiagonals.

0, 0, 1, 0, 2, 2, 0, 3, 1, 3, 0, 4, 2, 6, 4, 0, 5, 3, 1, 8, 5, 0, 6, 4, 2, 12, 7, 6, 0, 7, 5, 3, 1, 15, 3, 7, 0, 8, 6, 4, 2, 20, 14, 5, 8, 0, 9, 7, 5, 3, 1, 24, 13, 4, 9, 0, 10, 8, 6, 4, 2, 30, 23, 8, 18, 10, 0, 11, 9, 7, 5, 3, 1, 35, 22, 11, 20, 11, 0, 12, 10, 8, 6, 4, 2, 42, 34, 21, 10, 19, 12

Offset: 0

Alois P. Heinz, Mar 03 2015

Every column is a permutation of the nonnegative integers.

			Square array A(n,k) begins:
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, ...
   2,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
   3,  6,  1,  2,  3,  4,  5,  6,  7,  8,  9, ...
   4,  8, 12,  1,  2,  3,  4,  5,  6,  7,  8, ...
   5,  7, 15, 20,  1,  2,  3,  4,  5,  6,  7, ...
   6,  3, 14, 24, 30,  1,  2,  3,  4,  5,  6, ...
   7,  5, 13, 23, 35, 42,  1,  2,  3,  4,  5, ...
   8,  4,  8, 22, 34, 48, 56,  1,  2,  3,  4, ...
   9, 18, 11, 21, 33, 47, 63, 72,  1,  2,  3, ...
  10, 20, 10, 15, 32, 46, 62, 80, 90,  1,  2, ...

Columns k=2-16 give: A001477, A004488, A048647, A055115, A055116, A055117, A055118, A055119, A055120, A055121, A055122, A055123, A055124, A055125, A055126.

A:= proc(n, k) local t, r, i; t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+k^i *irem(k-irem(t, k, 't'), k)
      od; r
    end:
seq(seq(A(n, 2+d-n), n=0..d), d=0..14);

A(n,k)=fromdigits(apply(d->(k-d)%k, digits(n, k)), k); \\ Gheorghe Coserea, Apr 23 2018

A055115 Base-5 complement of n (write n in base 5, then replace each digit with its base-5 negative).

Original entry on oeis.org

0, 4, 3, 2, 1, 20, 24, 23, 22, 21, 15, 19, 18, 17, 16, 10, 14, 13, 12, 11, 5, 9, 8, 7, 6, 100, 104, 103, 102, 101, 120, 124, 123, 122, 121, 115, 119, 118, 117, 116, 110, 114, 113, 112, 111, 105, 109, 108, 107, 106, 75, 79, 78, 77, 76, 95, 99, 98, 97, 96, 90, 94, 93, 92

Offset: 0

Henry Bottomley, Apr 19 2000

Cf. A004488, A048647, A055115-A055126.

Column k=5 of A248813.

A055126 Base-16 complement of n (write n in base 16, then replace each digit with its base-16 negative).

Original entry on oeis.org

0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 240, 255, 254, 253, 252, 251, 250, 249, 248, 247, 246, 245, 244, 243, 242, 241, 224, 239, 238, 237, 236, 235, 234, 233, 232, 231, 230, 229, 228, 227, 226, 225, 208, 223, 222, 221, 220, 219, 218, 217, 216, 215

Offset: 0

Henry Bottomley, Apr 19 2000

Reinhard Zumkeller, Table of n, a(n) for n = 0..4095 (4095 = 16^3 - 1)
Index entries for sequences that are permutations of the natural numbers

Cf. A004488, A048647, A055115-A055126.

Column k=16 of A248813.

a055126 0 = 0
a055126 n = if d == 0 then 16 * a055126 n' else 16 * a055126 n' + 16 - d
            where (n', d) = divMod n 16
-- Reinhard Zumkeller, Mar 12 2014

A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 7, 12, 12, 7, 6, 0, 0, 7, 3, 15, 16, 15, 3, 7, 0, 0, 8, 5, 18, 11, 11, 18, 5, 8, 0, 0, 9, 4, 21, 24, 13, 24, 21, 4, 9, 0, 0, 10, 18, 24, 19, 21, 21, 19, 24, 18, 10, 0, 0, 11, 20, 27, 23, 26, 9, 26, 23, 27, 20, 11, 0, 0, 12, 19, 30, 36, 19, 15, 15, 19, 36, 30, 19, 12, 0

Offset: 0

Antti Karttunen, May 22 2019

			The array begins as:
  0,  0,  0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0, ...
  0,  1,  2,  3,  4,  5,  6,  7,  8,   9,  10,  11,  12, ...
  0,  2,  1,  6,  8,  7,  3,  5,  4,  18,  20,  19,  24, ...
  0,  3,  6,  9, 12, 15, 18, 21, 24,  27,  30,  33,  36, ...
  0,  4,  8, 12, 16, 11, 24, 19, 23,  36,  40,  44,  48, ...
  0,  5,  7, 15, 11, 13, 21, 26, 19,  45,  50,  52,  33, ...
  0,  6,  3, 18, 24, 21,  9, 15, 12,  54,  60,  57,  72, ...
  0,  7,  5, 21, 19, 26, 15, 13, 11,  63,  70,  68,  57, ...
  0,  8,  4, 24, 23, 19, 12, 11, 16,  72,  80,  76,  69, ...
  0,  9, 18, 27, 36, 45, 54, 63, 72,  81,  90,  99, 108, ...
  0, 10, 20, 30, 40, 50, 60, 70, 80,  90, 100,  83, 120, ...
  0, 11, 19, 33, 44, 52, 57, 68, 76,  99,  83,  91, 132, ...
  0, 12, 24, 36, 48, 33, 72, 57, 69, 108, 120, 132, 144, ...
  etc.
A(2,2) = 2*2 mod 3 = 1.

Cf. A169999 (the main diagonal).
Row/Column 0: A000004, Row/Column 1: A001477, Row/Column 2: A004488, Row/Column 3: A008585, Row/Column 4: A242399, Row/Column 9: A008591.
Cf. A007089, A207669, A207670, A263273, A325825.
Cf. A325821 (same table without the zero row and column).
Cf. A048720 (binary), A059692 (decimal), A004247 (full multiply).

up_to = 105;
A325820sq(b, c) = fromdigits(Vec(Pol(digits(b,3))*Pol(digits(c,3)))%3, 3);
A325820list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A325820sq(a-col,col))); (v); };
v325820 = A325820list(up_to);
A325820(n) = v325820[1+n];

A004489 Table of tersums m + n (answers written in base 10).

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

Offset: 0

N. J. A. Sloane

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

Alois P. Heinz, Antidiagonals d = 0..140, flattened

Similar to but different from A004481.
Main diagonal gives A004488.
Cf. A003987 (analogous sequence for base 2).

T:= proc(n, m) local t, h, r, i;
      t, h, r:= n, m, 0;
      for i from 0 while t>0 or h>0 do
        r:= r +3^i *irem(irem(t, 3, 't') +irem(h, 3, 'h'), 3)
      od; r
    end:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 07 2011

T[n_, m_] := Module[{t, h, r, i, remt, remh}, {t, h, r} = {n, m, 0}; For[i = 0, t>0 || h>0, i++, r = r + 3^i*Mod[({t, remt} = QuotientRemainder[t, 3 ]; remt) + ({h, remh} = QuotientRemainder[h, 3]; remh), 3]]; r]; Table[Table[T[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Jan 07 2014, translated from Maple *)

T(n,m) = fromdigits(Vec(Pol(digits(n,3)) + Pol(digits(m,3)))%3, 3); \\ Kevin Ryde, Apr 06 2021

def T(n, m):
  k, pow3 = 0, 1
  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([T(n, d-n) for d in range(14) for n in range(d+1)]) # Michael S. Branicky, May 04 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.

More terms from Larry Reeves (larryr(AT)acm.org), Jan 23 2001

A289831 a(n) = A289813(n) + A289814(n).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Jul 13 2017

The ones in the binary representation of a(n) correspond to the nonzero digits in the ternary representation of n; for example: ternary(42) = 1120 and binary(a(42)) = 1110 (a(42) = 14).
Each number k >= 0 appears 2^A000120(k) times.
a(A004488(n)) = a(n).

			The first values, alongside the ternary representation of n, and the binary representation of a(n), are:
n       a(n)    ternary(n)  binary(a(n))
--      ----    ----------  ------------
0       0       0           0
1       1       1           1
2       1       2           1
3       2       10          10
4       3       11          11
5       3       12          11
6       2       20          10
7       3       21          11
8       3       22          11
9       4       100         100
10      5       101         101
11      5       102         101
12      6       110         110
13      7       111         111
14      7       112         111
15      6       120         110
16      7       121         111
17      7       122         111
18      4       200         100
19      5       201         101
20      5       202         101
21      6       210         110
22      7       211         111
23      7       212         111
24      6       220         110
25      7       221         111
26      7       222         111

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A000120, A004488, A289813, A289814.

Table[FromDigits[Sign@ IntegerDigits[n, 3], 2], {n, 0, 100}] (* Indranil Ghosh, Aug 03 2017 *)

a(n) = my (d=digits(n,3)); fromdigits(vector(#d, i, sign(d[i])), 2)

from sympy.ntheory.factor_ import digits
from sympy import sign
def a(n):
    d=digits(n, 3)[1:]
    return int(''.join(str(sign(i)) for i in d), 2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Aug 03 2017

a(0) = 0.
a(3*n) = 2*a(n).
a(3*n + 1) = 2*a(n) + 1.
a(3*n + 2) = 2*a(n) + 1.

A244042 In ternary representation of n, replace 2's with 0's.

Original entry on oeis.org

0, 1, 0, 3, 4, 3, 0, 1, 0, 9, 10, 9, 12, 13, 12, 9, 10, 9, 0, 1, 0, 3, 4, 3, 0, 1, 0, 27, 28, 27, 30, 31, 30, 27, 28, 27, 36, 37, 36, 39, 40, 39, 36, 37, 36, 27, 28, 27, 30, 31, 30, 27, 28, 27, 0, 1, 0, 3, 4, 3, 0, 1, 0, 9, 10, 9, 12, 13, 12, 9, 10, 9

Offset: 0

Joonas Pohjonen, Jun 17 2014

			16 = 121_3, replacing 2 with 0 gives 101_3 = 10, so a(16) = 10.

Alois P. Heinz, Table of n, a(n) for n = 0..6560

Cf. A004488, A117966, A005836, A289813, A289814.

a:= proc(n) local t, r, i; t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i*(d-> `if`(d=2, 0, d))(irem(t, 3, 't'))
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 17 2014

Array[FromDigits[IntegerDigits[#, 3] /. 2 -> 0, 3] &, 72, 0] (* Michael De Vlieger, Mar 17 2018 *)

a(n) = my(d=digits(n, 3)); fromdigits(apply(x->(if (x==2, 0, x)), d), 3); \\ Michel Marcus, Jun 10 2017

from sympy.ntheory.factor_ import digits
def a(n):return int("".join(map(str, digits(n, 3)[1:])).replace('2', '0'), 3) # Indranil Ghosh, Jun 10 2017

a(n) = n - 2 * A005836(A289814(n) + 1) = A005836(A289813(n) + 1). - Andrey Zabolotskiy, Nov 11 2019

A140263 Permutation of nonnegative integers obtained by interleaving A117967 and A117968.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 19 2008, originally described in a posting at the SeqFan mailing list on Sep 15 2005

Inverse: A140264. Bisections: A117967 & A117968. a(n) = A140265(n+1)-1.

Cf. A004488, A117966.

from sympy import ceiling
from sympy.ntheory.factor_ import digits
def a004488(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3)
def a117968(n):
    if n==1: return 2
    if n%3==0: return 3*a117968(n/3)
    elif n%3==1: return 3*a117968((n - 1)/3) + 2
    else: return 3*a117968((n + 1)/3) + 1
def a117967(n): return 0 if n==0 else a117968(-n) if n<0 else a004488(a117968(n))
def a001057(n): return -(-1)**n*ceiling(n/2)
def a(n): return a117967(a001057(n)) # Indranil Ghosh, Jun 07 2017

a(n) = A117967(A001057(n)). (Assuming that the domain of A117967 is the whole Z line.)

A242399 Write n and 3n in ternary representation and add all trits modulo 3.

Original entry on oeis.org

0, 4, 8, 12, 16, 11, 24, 19, 23, 36, 40, 44, 48, 52, 47, 33, 28, 32, 72, 76, 80, 57, 61, 56, 69, 64, 68, 108, 112, 116, 120, 124, 119, 132, 127, 131, 144, 148, 152, 156, 160, 155, 141, 136, 140, 99, 103, 107, 84, 88, 83, 96, 91, 95, 216, 220, 224, 228, 232

Offset: 0

Reinhard Zumkeller, May 13 2014

			n = 25, 3*n = 75:
.  A007089(25) =  221
.  A007089(75) = 2210
.   add trits    ----
.    modulo 3    2101 = A007089(64), hence a(25) = 64.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system
Index entries for sequences related to carryless arithmetic

Cf. A004488, A007089, A008586, A030341, A048724, A242400, A242407.

Row / column 4 of A325820.

a242399 n = foldr (\t v -> 3 * v + t) 0 $
                  map (flip mod 3) $ zipWith (+) ([0] ++ ts) (ts ++ [0])
            where ts = a030341_row n

a(n) <= 4*n; a(m) = 4*m iff m is a term of A242407.

a(n) = A008586(n) - A242400(n).

A246207 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, a(2n) = A117968(a(n)), a(2n+1) = A117967(1+a(n)).

Original entry on oeis.org

0, 1, 2, 5, 7, 3, 22, 15, 23, 11, 6, 4, 71, 35, 66, 52, 58, 33, 25, 12, 21, 16, 8, 17, 172, 99, 73, 36, 213, 148, 194, 137, 197, 152, 75, 43, 59, 29, 24, 13, 69, 49, 68, 47, 19, 9, 64, 45, 587, 419, 225, 127, 173, 104, 72, 37, 516, 304, 620, 431, 643, 447, 601, 462, 640, 441, 577, 423, 177, 103, 203, 155, 211, 150, 61, 30, 57, 34, 26, 53

Offset: 0

Antti Karttunen, Aug 19 2014

This is an instance of entanglement permutation, where complementary pair A005843/A005408 (even and odd numbers respectively) is entangled with complementary pair A117968/A117967 (negative and positive part of inverse of balanced ternary enumeration of integers, respectively), with a(0) set to 0 and a(1) set to 1.

Thus this shares with A140263 the property that after a(0)=0, the even positions contain only terms of A117968 and the odd positions contain only terms of A117967.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Pin & logarithmic scatter plots computed for range 0..2047 only (computed with OEIS "graph" command)
Index entries for sequences that are permutations of the natural numbers

Inverse: A246208.
Related permutations: A140263, A054429, A246209, A246211.
Cf. A117967, A117968.

from sympy.ntheory.factor_ import digits
def a004488(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3)
def a117968(n):
    if n==1: return 2
    if n%3==0: return 3*a117968(n/3)
    elif n%3==1: return 3*a117968((n - 1)/3) + 2
    else: return 3*a117968((n + 1)/3) + 1
def a117967(n): return 0 if n==0 else a117968(-n) if n<0 else a004488(a117968(n))
def a(n): return n if n<2 else a117968(a(n/2)) if n%2==0 else a117967(1 + a((n - 1)/2)) # Indranil Ghosh, Jun 07 2017

As a composition of related permutations:
a(n) = A246209(A054429(n)).
a(n) = A246211(A246209(n)).

cp's OEIS Frontend

A248813 A(n,k) is the base-k complement of n; square array A(n,k), n>=0, k>=2, read by antidiagonals.

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A055115 Base-5 complement of n (write n in base 5, then replace each digit with its base-5 negative).

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A055126 Base-16 complement of n (write n in base 16, then replace each digit with its base-16 negative).

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A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

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A004489 Table of tersums m + n (answers written in base 10).

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A289831 a(n) = A289813(n) + A289814(n).

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A244042 In ternary representation of n, replace 2's with 0's.

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A140263 Permutation of nonnegative integers obtained by interleaving A117967 and A117968.

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