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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.
+20
7
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
COMMENTS
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.
EXAMPLE
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.
PROG
(PARI) See Links section.
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.
+10
12
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
COMMENTS
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]
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 + 2*v) = (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.
+10
3
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
COMMENTS
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.
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 \
PROG
(PARI) See Links section.
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.
+10
2
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
COMMENTS
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)).
EXAMPLE
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
MATHEMATICA
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 *)
PROG
(PARI) a(n, p=[4, 5, 6, 1, 2, 3]) = fromdigits(apply(d -> if (d, p[d], 0), digits(n, #p+1)), #p+1)
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.
+10
2
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
COMMENTS
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.
FORMULA
a(2^(k+1)) = A077966(k) for any k >= 0.
PROG
(PARI) See Links section.
CROSSREFS
See A334493 for a similar sequence.
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 + 2*v) = (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.
+10
2
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
COMMENTS
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.
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 \
PROG
(PARI) See Links section.
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