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A335380
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a(n) is the X-coordinate of the n-th point of the Kochawave curve; sequence A335381 gives Y-coordinates.
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2
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0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 9, 12, 12, 13, 16, 14, 15, 16, 16, 17, 18, 18, 19, 18, 18, 19, 22, 20, 21, 20, 18, 19, 18, 18, 19, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:
Y
/
/
0 ---- X
The Kochawave curve is a variant of the Koch curve that can be built by successively applying the following substitution to an initial vector (1, 0):
.+ C
.../
... /
... /
+------>+. +------>+
A B D E
- the points A, B, D and E are aligned and equally spaced,
- the points D, C and E form an equilateral triangle
(for the Koch curve, the points B, C and D form an equilateral triangle).
The distance between two consecutive points is related to A160381:
- for any n >= 0, let z(n) = a(n) + A335381(n) * exp(i*Pi/3) (where i denotes the imaginary unit),
- the square of the distance from z(n) to z(n+1) is 3^A160381(n).
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LINKS
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FORMULA
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a(4^k) = 3^k for any k >= 0.
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EXAMPLE
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The Kochawave curve starts (on a hexagonal lattice) as follows:
. . . . . . + . . .
/|6
/ |
/ |
. . . . . . | . .+ . .
/ | .../ 8
/ | ... /
/ | ... /
. . . . . . +. + . .
/ 7 |9
/ |
/ |
. . .+ . .+ . +11 | . .+ .
.../ 2 ... 5 / \ | .../ 14
... / ... / \ | ... /
... / ... / \| ... /
+-------+. +-------+. . . +-------+. +-------+
0 1 3 4 12 13 10 15 16
- hence a(8) = a(9) = a(11) = a(12) = 6.
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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