A060374 -id:A060374 - cp's OEIS frontend

A060372 p(n), positive part of n when n=p(n)-q(n) with p(n), q(n), p(n)+q(n) in A005836, integers written without 2 in base 3.

0, 1, 3, 3, 4, 9, 9, 10, 9, 9, 10, 12, 12, 13, 27, 27, 28, 27, 27, 28, 30, 30, 31, 27, 27, 28, 27, 27, 28, 30, 30, 31, 36, 36, 37, 36, 36, 37, 39, 39, 40, 81, 81, 82, 81, 81, 82, 84, 84, 85, 81, 81, 82, 81, 81, 82, 84, 84, 85, 90, 90, 91, 90, 90, 91, 93, 93, 94, 81, 81, 82, 81

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Apr 02 2001

The graphs of p(n), q(n) are fractals; the graph of p(n)+q(n) is Sierpiński-like.

			Example: 14=27-13=3^3 -(3^0+3^1+3^2), 16=28-12=3^3+3^0 -(3^1+3^2), 20=30-10=3^3+3^1 -(3^0+3^2); 27+13=28+12=30+10=40; 10,12,13, 27, 28, 30 are written without 2 in base 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A005836, A060373, A060374.

a060372 n = (a060374 n + n) `div` 2  -- Reinhard Zumkeller, Jun 09 2012

A060373 q(n), negative part of n when n=p(n)-q(n) with p(n), q(n), p(n)+q(n) in A005836, integers written without 2 in base 3.

Original entry on oeis.org

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

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Apr 02 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A005836, A060372, A060374.

a060373 n = a060372 n - n  -- Reinhard Zumkeller, Jun 09 2012

A343316 Array T(n, k), n, k > 0, read by antidiagonals; the balanced ternary representation of T(n, k) is obtained by multiplying componentwise the digits in the balanced ternary representations of n and of k.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, -1, -1, 0, 0, 0, 4, 0, 0, 0, 1, 3, 3, 1, 0, 0, -1, 2, 3, 2, -1, 0, 0, 0, -2, 3, 3, -2, 0, 0, 0, 1, -3, -3, 4, -3, -3, 1, 0, 0, -1, -4, -3, -4, -4, -3, -4, -1, 0, 0, 0, 1, -3, -3, 13, -3, -3, 1, 0, 0, 0, 1, 0, 0, -2, 12, 12, -2, 0, 0, 1, 0

Offset: 0

Rémy Sigrist, Apr 11 2021

For any k >= 0, n -> T(n, k) is 3^A134021(k)-periodic.

The zeros of the table form a Vicsek fractal (see illustration in Links section).

			Array T(n, k) begins:
  n\k|  0   1   2   3   4   5   6   7   8  9  10  11  12
  ---+--------------------------------------------------
    0|  0   0   0   0   0   0   0   0   0  0   0   0   0
    1|  0   1  -1   0   1  -1   0   1  -1  0   1  -1   0
    2|  0  -1   4   3   2  -2  -3  -4   1  0  -1   4   3
    3|  0   0   3   3   3  -3  -3  -3   0  0   0   3   3
    4|  0   1   2   3   4  -4  -3  -2  -1  0   1   2   3
    5|  0  -1  -2  -3  -4  13  12  11  10  9   8   7   6
    6|  0   0  -3  -3  -3  12  12  12   9  9   9   6   6
    7|  0   1  -4  -3  -2  11  12  13   8  9  10   5   6
    8|  0  -1   1   0  -1  10   9   8  10  9   8  10   9
    9|  0   0   0   0   0   9   9   9   9  9   9   9   9
   10|  0   1  -1   0   1   8   9  10   8  9  10   8   9
   11|  0  -1   4   3   2   7   6   5  10  9   8  13  12
   12|  0   0   3   3   3   6   6   6   9  9   9  12  12
Array T(n, k) begins in balanced ternary notation (with "T" instead of digits "-1"):
  n\k|  0  1  1T  10  11  1TT  1T0  1T1  10T  100  101  11T  110
  ---+----------------------------------------------------------
    0|  0  0   0   0   0    0    0    0    0    0    0    0    0
    1|  0  1   T   0   1    T    0    1    T    0    1    T    0
   1T|  0  T  11  10  1T   T1   T0   TT    1    0    T   11   10
   10|  0  0  10  10  10   T0   T0   T0    0    0    0   10   10
   11|  0  1  1T  10  11   TT   T0   T1    T    0    1   1T   10
  1TT|  0  T  T1  T0  TT  111  110  11T  101  100  10T  1T1  1T0
  1T0|  0  0  T0  T0  T0  110  110  110  100  100  100  1T0  1T0
  1T1|  0  1  TT  T0  T1  11T  110  111  10T  100  101  1TT  1T0
  10T|  0  T   1   0   T  101  100  10T  101  100  10T  101  100
  100|  0  0   0   0   0  100  100  100  100  100  100  100  100
  101|  0  1   T   0   1  10T  100  101  10T  100  101  10T  100
  11T|  0  T  11  10  1T  1T1  1T0  1TT  101  100  10T  111  110
  110|  0  0  10  10  10  1T0  1T0  1T0  100  100  100  110  110

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 3^6 (where the color denotes the sign of T(n, k): red for positive values, blue for negative values, white for zeros)
Wikipedia, Viczek fractal

Cf. A059095, A060374, A102283, A134021, A343317.

T(n,k) = { if (n==0 || k==0, return (0), my (d=centerlift(Mod(n,3)), t=centerlift(Mod(k,3))); 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) = 0.
T(n, 1) = A102283(n).
T(n, n) = A060374(n).

cp's OEIS Frontend

A060372 p(n), positive part of n when n=p(n)-q(n) with p(n), q(n), p(n)+q(n) in A005836, integers written without 2 in base 3.

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A060373 q(n), negative part of n when n=p(n)-q(n) with p(n), q(n), p(n)+q(n) in A005836, integers written without 2 in base 3.

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A343316 Array T(n, k), n, k > 0, read by antidiagonals; the balanced ternary representation of T(n, k) is obtained by multiplying componentwise the digits in the balanced ternary representations of n and of k.

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