A334493 -id:A334493 - cp's OEIS frontend

A334492 a(n) is the "real" part of f(n) = Sum_{k>=0, d_k>0} (1+w)^(d_k-1) * (3+w)^k where Sum_{k>=0} d_k * 7^k is the base 7 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A334493 gives "w" parts.

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

Offset: 0

Rémy Sigrist, May 03 2020

For any Eisenstein integer z = u + v*w (where u and v are integers), we call u the "real" part of z and v the "w" part of z.
This sequence has connections with A316657; here we work with Eisenstein integers, there with Gaussian integers.
It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.

			The following diagram depicts f(n) for n = 0..13:
            "w" axis
                \
           .     .     .     .     .     .     .     .
                  \              10     9
                   \
        .     .     .     .     .     .     .     .
                   3 \   2    11     7     8
                      \
           ._____._____._____._____._____._____._____. "real" axis
                4     0 \   1    12    13
                         \
        .     .     .     .     .     .     .     .
                   5     6 \
- f(9) = 4 + 2*w, hence a(9) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..16806
Rémy Sigrist, Colored representation of f(n) for n = 0..7^7-1 in a hexagonal lattice (where the hue is function of n)
Rémy Sigrist, PARI program for A334492
Wikipedia, Eisenstein integer

Cf. A307013 (equivalent coordinate for a counterclockwise spiral), A316657, A334493.

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A307012 Second coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. First and third coordinates are given in A307011 and A345978.

Original entry on oeis.org

0, 0, 1, 1, 0, -1, -1, -1, 0, 1, 2, 2, 2, 1, 0, -1, -2, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 3, 2, 1, 0, -1, -2, -3, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4, 4, 4, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 4

Offset: 0

Hugo Pfoertner, Mar 19 2019

The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0. a(n) is the signed distance from spiral point n to the axis that passes through point 1. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 2 has positive distance. - Peter Munn, Jul 13 2021

We can use this coordinate with the first coordinate to form an oblique coordinate system, in which each coordinate maps to an oblique coordinate vector parallel to the axis along which the other coordinate is 0. See the figure with nonperpendicular axes in the Barile link. When both of these coordinates are positive, the oblique coordinate vectors make a 60-degree angle with each other. [Made more specific by Peter Munn, Jul 19 2021]

Hugo Pfoertner, Table of n, a(n) for n = 0..10150
Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
HandWiki, Hexagonal Lattice.
Peter Munn, Illustration of signed distance of spiral points.
Wikipedia, Signed distance function.
Index entries for sequences related to coordinates of 2D curves

Cf. A307011, A307013, A328818, A334493, A345435, A345978.

Name revised by Peter Munn, Jul 08 2021

A348917 a(n) is the "w" part of f(n) = Sum_{k >= 0} g(d_k) * (4 + w)^k where g(0) = 0 and g(1 + u + 2v) = (2 + w)^u (1 + w)^v for any u = 0..1 and v = 0..5, Sum_{k >= 0} d_k * 13^k is the base-13 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A348916 gives "real" parts.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Nov 03 2021

For any Eisenstein integer z = u + v*w (where u and v are integers), we call u the "real" part of z and v the "w" part of z.
This sequence combines features of A334493 and of A348653.
It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.
The following diagram depicts g(d) for d = 0..12:
           "w" axis
               \
                .     .
                 \   4
                  \
             .     .     .     .
            6     5 \   3     2
                     \
          .___._____.___._____._ "real" axis
               7     0 \   1
                        \
             .     .     .     .
            8     9    11 \  12
                           \
                      .     .
                    10       \

Rémy Sigrist, Table of n, a(n) for n = 0..2196
Joerg Arndt, Representation of a similar construction
Rémy Sigrist, Colored representation of f for n = 0..13^5-1 in the complex plane (the hue is function of n)
Rémy Sigrist, PARI program for A348917
Wikipedia, Eisenstein integer

Cf. A334493, A348653, A348916.

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A348355 The base-7 expansion of a(n) is obtained by replacing 1's, 2's, 3's, 4's, 5's and 6's by 4's, 5's, 6's, 1's, 2's and 3's, respectively, in the base-7 expansion of n.

Original entry on oeis.org

0, 4, 5, 6, 1, 2, 3, 28, 32, 33, 34, 29, 30, 31, 35, 39, 40, 41, 36, 37, 38, 42, 46, 47, 48, 43, 44, 45, 7, 11, 12, 13, 8, 9, 10, 14, 18, 19, 20, 15, 16, 17, 21, 25, 26, 27, 22, 23, 24, 196, 200, 201, 202, 197, 198, 199, 224, 228, 229, 230, 225, 226, 227, 231

Offset: 0

Rémy Sigrist, Oct 14 2021

This sequence is a self-inverse permutation of the nonnegative integers.
It is possible to build a similar sequence for any fixed base b > 1 and any permutation p of {1, ..., b-1}.
This sequence is interesting as it satisfies f(a(n)) = -f(n), where f(n) = (A334492(n), A334493(n)).

			The first terms, in decimal and in base 7, are:
  n   a(n)  s(n)  s(a(n))
  --  ----  ----  -------
   0     0     0        0
   1     4     1        4
   2     5     2        5
   3     6     3        6
   4     1     4        1
   5     2     5        2
   6     3     6        3
   7    28    10       40
   8    32    11       44
   9    33    12       45
  10    34    13       46

See A004488, A048647 and A348354 for similar sequences.

Cf. A334492, A334493.

a[n_] := With[{d = {0, 4, 5, 6, 1, 2, 3}}, FromDigits[d[[IntegerDigits[n, 7] + 1]], 7]]; Array[a, 64, 0] (* Amiram Eldar, Oct 16 2021 *)

a(n, p=[4,5,6,1,2,3]) = fromdigits(apply(d -> if (d, p[d], 0), digits(n, #p+1)), #p+1)

A348911 a(n) is the "w" part of f(n) = Sum_{k>=0, d_k>0} w^(d_k-1) * (-2)^k where Sum_{k>=0} d_k * 4^k is the base-4 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A348910 gives "real" parts.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Nov 03 2021

For any Eisenstein integer z = u + v*w (where u and v are integers), we call u the "real" part of z and v the "w" part of z.

The function f defines a bijection from the nonnegative integers to the Eisenstein integers.

Rémy Sigrist, Table of n, a(n) for n = 0..16383
Rémy Sigrist, Colored representation of f(n) for n = 0..4^10-1 in a hexagonal lattice (where the hue is function of n)
Rémy Sigrist, PARI program for A348911
Gary Teachout, Fractal Space Filling Curves 2002
Wikipedia, Eisenstein integer

See A334493 for a similar sequence.

Cf. A077966, A348910.

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a(2^(k+1)) = A077966(k) for any k >= 0.

A348921 a(n) is the "w" part of f(n) = Sum_{k >= 0} g(d_k) * (4 + w)^k where g(0) = 0 and g(1 + u + 2v) = (1 + u) (1 + w)^v for any u = 0..1 and v = 0..5, Sum_{k >= 0} d_k * 13^k is the base-13 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A348920 gives "real" parts.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Nov 04 2021

For any Eisenstein integer z = u + v*w (where u and v are integers), we call u the "real" part of z and v the "w" part of z.
This sequence is a variant of A334493 and of A348917.
It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.
The following diagram depicts g(d) for d = 0..12:
           "w" axis
               \
                .     .     .
               6 \         4
                  \
                   .     .
                  5 \   3
                     \
          .___._____.___._____._ "real" axis
         8     7     0 \   1     2
                        \
                   .     .
                  9    11 \
                           \
                .     .     .
              10          12 \

Rémy Sigrist, Table of n, a(n) for n = 0..2196
Joerg Arndt, Representation of a similar construction
Rémy Sigrist, Colored representation of f for n = 0..13^5-1 in the complex plane (the hue is function of n)
Rémy Sigrist, PARI program for A348921
Wikipedia, Eisenstein integer

Cf. A334493, A348917, A348920.

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A334492 a(n) is the "real" part of f(n) = Sum_{k>=0, d_k>0} (1+w)^(d_k-1) * (3+w)^k where Sum_{k>=0} d_k * 7^k is the base 7 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A334493 gives "w" parts.

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A307012 Second coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. First and third coordinates are given in A307011 and A345978.

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A348355 The base-7 expansion of a(n) is obtained by replacing 1's, 2's, 3's, 4's, 5's and 6's by 4's, 5's, 6's, 1's, 2's and 3's, respectively, in the base-7 expansion of n.

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A348911 a(n) is the "w" part of f(n) = Sum_{k>=0, d_k>0} w^(d_k-1) * (-2)^k where Sum_{k>=0} d_k * 4^k is the base-4 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A348910 gives "real" parts.

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