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a(n) = A059253(n) + A283316(n+1).

Rémy Sigrist, <a href="/A341121/b341121.txt">Table of n, a(n) for n = 0..16384</a>

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a(2*n) = A341019(n).

a(4*n) = 2*A341120(n).

a(16*n) = 4*a(n).

A059252(n) = (a(4*n+2)-1)/2.

Points n and their locations X=A341120(n), Y=a(n) begin as follows. n=7 and n=9 are both at X=3,Y=2, and n=11,n=31 both at X=3,Y=4.

| |

4 | 16---17 12--11,31

| | | |

3 | 15---14---13 10

| |

2 | 8---7,9

| |

1 | 1----2----3 6

| | | |

Y=0 | 0 4----5

+--------------------

X=0 1 2 3

Cf. A059252, A341019, A341120.

<a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

<a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

allocated for Rémy Sigrist

a(n) is the Y-coordinate of the n-th point of the space filling curve C defined in Comments section; A341120 gives X-coordinates.

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

0,8

We define the family {C_n, n >= 0}, as follows:

- C_0 corresponds to the points (0, 0), (0, 1), (1, 1), (2, 1) and (2, 0), in that order:

+---+---+

| |

+ +

O

- for any n >= 0, C_{n+1} is obtained by arranging 4 copies of C_n as follows:

+ . . . + . . . +

. B . B .

+ . . . + . . .

. B . .A C.A C.

. . --> + . . . + . . . +

.A C. .C . A.

+ . . . + . B.B .

O .A . C.

+ . . . + . . . +

O

- the space filling curve C is the limit of C_{2*n} as n tends to infinity.

F. M. Dekking, <a href="http://dx.doi.org/10.1016/0001-8708(82)90066-4">Recurrent Sets</a>, Advances in Mathematics, vol. 44, no. 1, 1982.

<a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

Rémy Sigrist, <a href="/A341121/a341121.gp.txt">PARI program for A341121</a>

(PARI) See Links section.

Cf. A341120.

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Kevin Ryde and Rémy Sigrist, Feb 05 2021

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allocated for Rémy Sigrist

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