A332497 -id:A332497 - cp's OEIS frontend

A332498 a(n) = y(w+1) where y(0) = 0 and y(k+1) = 2^(k+1)-1-y(k) (resp. y(k)) when d_k = 2 (resp. d_k <> 2) and Sum_{k=0..w} d_k*3^k is the ternary representation of n. Sequence A332497 gives corresponding x's.

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

Offset: 0

Rémy Sigrist, Feb 14 2020

			For n = 42:
- the ternary representation of 42 is "1120",
- x(0) = 0,
- x(1) = x(0) = 0 (as d_0 = 0),
- x(2) = 2^2-1 - x(1) = 3 (as d_1 = 2),
- x(3) = x(2) = 3 (as d_2 = 1 <> 2),
- x(4) = x(3) = 3 (as d_3 = 1 <> 2),
- hence a(42) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Wikipedia, T-square (fractal)

Cf. A005836, A332497 (corresponding x's and additional comments).

a(n) = { my (y=0, k=1); while (n, if (n%3==2, y=2^k-1-y); n\=3; k++); y }

a(n) = 0 iff n belongs to A005836.

A332412 a(n) is the real part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332413 gives imaginary parts.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Feb 12 2020

The representation of {f(n)} corresponds to the cross form of the Vicsek fractal.

As a set, {f(n)} corresponds to the Gaussian integers whose real and imaginary parts have not simultaneously a nonzero digit at the same place in their balanced ternary representations.

			For n = 103:
- 103 = 4*5^2 + 3*5^0,
- so f(123) = 3^2 * i^(4-1) + 3^0 * i^(3-1) = -1 - 9*i,
- and a(n) = -1.

Rémy Sigrist, Table of n, a(n) for n = 0..15624
Rémy Sigrist, Colored representation of f(n) for n = 0..5^6-1 in the complex plan (where the hue is function of n)
Wikipedia, Vicsek fractal

See A332497 for a similar sequence.

Cf. A031219, A289813, A332413 (imaginary parts).

a(n) = { my (d=Vecrev(digits(n,5))); real(sum (k=1, #d, if (d[k], 3^(k-1)*I^(d[k]-1), 0))) }

a(n) = 0 iff the n-th row of A031219 has only even terms.
a(5*n)   = 3*a(n).
a(5*n+1) = 3*a(n) + 1.
a(5*n+2) = 3*a(n).
a(5*n+3) = 3*a(n) - 1.
a(5*n+4) = 3*a(n).

A339601 Starting from x_0 = n, iterate by dividing with 3 (discarding any remainder), until zero is reached: x_1 = floor(x_0/3), x_2 = floor(x_1/3), etc. Then a(n) = Sum_{i=0..} (x_i AND 2^i), where AND is bitwise-and.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3

Offset: 0

Antti Karttunen, Dec 09 2020

Index entries for sequences related to binary expansion of n

Cf. A004198, A037095.

Cf. also A332497.

Array[Total@ MapIndexed[BitAnd[2^First[#2 - 1], #1] &, NestWhileList[Floor[#/3] &, #, # > 0 &]] &, 106, 0] (* Michael De Vlieger, Dec 10 2020 *)

A339601(n) = { my(i=0, s=0); while(n, s += bitand(2^i,n); i++; n \= 3); (s); };

A339601(n) = { my(m=1, s=0); while(n>=m, s += bitand(m,n); m <<= 1; n \= 3); (s); };

A363710 a(n) is the number of pairs of nonnegative integers (x, y) such that x + y = n and A003188(x) AND A003188(y) = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 2, 4, 8, 6, 4, 6, 4, 2, 2, 4, 8, 10, 8, 12, 12, 6, 4, 6, 12, 8, 4, 6, 4, 2, 2, 4, 8, 10, 8, 16, 20, 10, 8, 12, 24, 20, 12, 16, 12, 6, 4, 6, 12, 16, 12, 16, 16, 8, 4, 6, 12, 8, 4, 6, 4, 2, 2, 4, 8, 10, 8, 16, 20, 10, 8, 16, 32, 28, 20, 28

Offset: 0

Rémy Sigrist, Jun 17 2023

Equivalently, a(n) is the number of k >= 0 such that A332497(k) + A332498(k) = n.

The set of pairs of nonnegative integers (x, y) such that A003188(x) AND A003188(y) = 0 is related to the T-square fractal (see illustration in Links section).

			For n = 8:
- we have:
  k  A332497(8-k)  A332497(k)  A332497(8-k) AND A332497(k)
  -  ------------  ----------  ---------------------------
  0            12           0                            0
  1             4           1                            0
  2             5           3                            1
  3             7           2                            2
  4             6           6                            6
  5             2           7                            2
  6             3           5                            1
  7             1           4                            0
  8             0          12                            0
- so a(8) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of (x, y) such that x, y < 2^10 and A003188(x) AND A003188(y) = 0
Wikipedia, T-square (fractal)

Cf. A003188, A075427, A332497, A332498.

a(n) = 2*sum(k=0, n\2, bitand(bitxor(n-k, (n-k)\2), bitxor(k, k\2))==0) - (n==0)

A363710=lambda n: sum(map(lambda k: not (k^k>>1)&(n-k^n-k>>1),range(n+1>>1)))<<1 if n else 1 # Natalia L. Skirrow, Jun 22 2023

a(n) = 2 iff n belongs to A075427.

cp's OEIS Frontend

A332498 a(n) = y(w+1) where y(0) = 0 and y(k+1) = 2^(k+1)-1-y(k) (resp. y(k)) when d_k = 2 (resp. d_k <> 2) and Sum_{k=0..w} d_k*3^k is the ternary representation of n. Sequence A332497 gives corresponding x's.

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A332412 a(n) is the real part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332413 gives imaginary parts.

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A339601 Starting from x_0 = n, iterate by dividing with 3 (discarding any remainder), until zero is reached: x_1 = floor(x_0/3), x_2 = floor(x_1/3), etc. Then a(n) = Sum_{i=0..} (x_i AND 2^i), where AND is bitwise-and.

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A363710 a(n) is the number of pairs of nonnegative integers (x, y) such that x + y = n and A003188(x) AND A003188(y) = 0 (where AND denotes the bitwise AND operator).

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