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A341019
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a(n) is the Y-coordinate of the n-th point of the space filling curve M defined in Comments section; A341018 gives X-coordinates.
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3
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0, 1, 0, 1, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 6, 5, 4, 5, 6, 7, 6, 7, 6, 5, 4, 3, 4, 3, 2, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 4, 3, 4, 5, 4, 5, 6, 7, 8, 7, 8, 7, 6, 5, 6, 5, 6, 7, 8, 9, 10, 11, 10, 11, 10, 9, 8, 9, 8, 9, 10, 11, 12, 11, 12, 13, 12
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OFFSET
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0,5
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COMMENTS
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We define the family {M_n, n >= 0}, as follows:
- M_0 corresponds to the points (0, 0), (1, 1) and (2, 0), in that order:
+
/ \
/ \
+ +
O
- for any n >= 0, M_{n+1} is obtained by arranging 4 copies of M_n as follows:
+ . . . + . . . +
. B . B .
+ . . . + . . .
. B . .A C.A C.
. . --> + . . . + . . . +
.A C. .C . A.
+ . . . + . B.B .
O .A . C.
+ . . . + . . . +
O
- for any n >= 0, M_n has A087289(n) points,
- the space filling curve M is the limit of M_{2*n} as n tends to infinity.
The odd bisection of M is similar to a Hilbert's Hamiltonian walk (hence the connection with A059252).
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LINKS
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F. M. Dekking, Recurrent Sets, Advances in Mathematics, vol. 44, no. 1, 1982.
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FORMULA
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a(16*n) = 4*a(n).
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EXAMPLE
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The curve M starts as follows:
11+ 13+ +19 +21
/ \ / \ / \ / \
10+ 12+ 14+18 +20 +22
\ / \ /
9+ 15+ +17 +23
/ \ / \
8+ 6+ + +26 +24
\ / \ 16 / \ /
7+ 5+ +27 +25
/ \
4+ +28
\ /
1+ 3+ +29 +31
/ \ / \ / \
0+ 2+ +30 +32
- so a(0) = a(2) = a(30) = a(32) = 0,
a(1) = a(3) = a(29) = a(31) = 1.
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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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