A285698 Number of super perfect rhythmic tilings of [0,4n-1] with quadruples.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 110, 0, 0, 0, 0, 0, 0

Offset

1,24

Comments

A super perfect tiling of the line with quadruples consists of n groups of four evenly spaced points, each group having a different common interval such that all points of the line are covered, and such that all intervals are inferior or equal to n (thus, each interval belongs to [1..n]).

Links

Table of n, a(n) for n=1..31.

Formula

For n > 1, a(n) = A284757(n)*2 because A284757 ignores reflected solutions. - Fausto A. C. Cariboni, May 20 2017

a(n) = 0 if (n mod 8) not in {0, 1}, - Max Alekseyev, Sep 28 2023

Example

For n = 24, there are 20 tilings.
One is: (0,3,6,9), (1,7,13,19), (2,14,26,38), (4,11,18,25), (5,29,53,77), (8,12,16,20), (10,27,44,61), (15,37,59,81), (17,33,49,65), (21,36,51,66), (22,43,64,85), (23,34,45,56), (24,47,70,93), (28,41,54,67), (30,39,48,57), (31,50,69,88), (32,46,60,74), (35,55,75,95), (40,58,76,94), (42,52,62,72), (63,71,79,87), (68,73,78,83), (80,82,84,86), (89,90,91,92)
It can also be represented as (where each number is the interval of the group the point of the line belongs to):
3 6 12 3 7 24 3 6 4 3 17 7 4 6 12 22 4 16 7 6 4 15 21 11 23 7 12 17 13 24 9 19 14 16 11 20 15 22 12 9 18 13 10 21 17 11 14 23 9 16 19 15 10 24 13 20 11 9 18 22 14 17 10 8 21 16 15 13 5 19 23 8 10 5 14 20 18 24 5 8 2 22 2 5 2 21 2 8 19 1 1 1 1 23 18 20
Another one is: (0,23,46,69), (1,25,49,73), (2,4,6,8), (3,7,11,15), (5,26,47,68), (9,14,19,24), (10,27,44,61), (12,20,28,36), (13,35,57,79), (16,34,52,70), (17,31,45,59), (18,33,48,63), (21,32,43,54), (22,42,62,82), (29,41,53,65), (30,40,50,60), (37,56,75,94), (38,51,64,77), (39,55,71,87), (58,67,76,85), (66,72,78,84), (74,81,88,95), (80,83,86,89), (90,91,92,93)
It can also be represented as:
23 24 2 4 2 21 2 4 2 5 17 4 8 22 5 4 18 14 15 5 8 11 20 23 5 24 21 17 8 12 10 14 11 15 18 22 8 19 13 16 10 12 20 11 17 14 23 21 15 24 10 13 18 12 11 16 19 22 9 14 10 17 20 15 13 12 6 9 21 23 18 16 6 24 7 19 9 13 6 22 3 7 20 3 6 9 3 16 7 3 1 1 1 1 19 7

Crossrefs

Cf. A285527, A261517.

Sequence in context: A202957 A232586 A189437 * A109854 A008383 A024192

Adjacent sequences: A285695 A285696 A285697 * A285699 A285700 A285701

Keyword

nonn,more

Author

Tony Reix, Apr 25 2017

Extensions

a(27)-a(31) from Max Alekseyev, Sep 24 2023

Status

approved