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