A348712 Square array read by falling antidiagonals: T(n,k) is the number of bounded regions formed by the Lissajous curve x=cos(n*t), y=sin(k*t).

1, 2, 0, 3, 1, 3, 4, 1, 8, 0, 5, 2, 1, 0, 5, 6, 2, 18, 3, 14, 0, 7, 3, 23, 1, 23, 3, 7, 8, 3, 2, 6, 32, 0, 20, 0, 9, 4, 33, 1, 1, 8, 33, 0, 9, 10, 4, 38, 9, 50, 10, 46, 7, 26, 0, 11, 5, 3, 2, 59, 1, 59, 0, 3, 5, 11, 12, 5, 48, 12, 68, 15, 72, 14, 60, 9, 32, 0

Offset

1,2

Links

Table of n, a(n) for n=1..78.

Wikipedia, Lissajous curve

Formula

T(n,k) = (n-1)*(k-1) + n*k when n is odd and GCD(n,k) = 1.

T(n,k) = (n-1)*(k-1)/2 when n is even and GCD(n,k) = 1.

T(n,k) = T(n/m,k/m) when GCD(n,k) = m.

Example

Array begins:
+-----+---------------------------------------------------------------+
| n\k |  1    2    3    4    5    6    7    8    9   10   11   12  .. |
+-----+---------------------------------------------------------------+
|  1  |  1    2    3    4    5    6    7    8    9   10   11   12  .. |
|  2  |  0    1    1    2    2    3    3    4    4    5    5    6  .. |
|  3  |  3    8    1   18   23    2   33   38    3   48   53    4  .. |
|  4  |  0    0    3    1    6    1    9    2   12    2   15    3  .. |
|  5  |  5   14   23   32    1   50   59   68   77    2   95  104  .. |
|  6  |  0    3    0    8   10    1   15   18    1   23   25    2  .. |
|  7  |  7   20   33   46   59   72    1   98  111  124  137  150  .. |
|  8  |  0    0    7    0   14    3   21    1   28    6   35    1  .. |
|  9  |  9   26    3   60   77    8  111  128    1  162  179   18  .. |
| 10  |  0    5    9   14    0   23   27   32   36    1   45   50  .. |
| 11  | 11   32   53   74   95  116  137  158  179  200    1  242  .. |
| 12  |  0    0    0    3   22    0   33    8    3   10   55    1  .. |
| ..  |  ..  ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..  .. |
+---------------------------------------------------------------------+

Maple

T := proc(n, k) option remember; igcd(n, k); if % = 1 then (n-1)*(k-1);
ifelse(n::even, % / 2, % + n*k) else T(n / %, k / %) fi end:
seq(seq(T(k, n - k + 1), k = 1..n), n = 1..12); # Peter Luschny, Oct 31 2021

Mathematica

T[n_, k_] := T[n, k] = With[{m = GCD[n, k]}, Which[OddQ[n] && m == 1, (n-1)*(k-1)+n*k, EvenQ[n] && m == 1, (n-1)*(k-1)/2, True, T[n/m, k/m]]];
Table[Table[T[k, n - k + 1], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Oct 28 2023 *)

Crossrefs

Cf. A007678, A300153.

Sequence in context: A212217 A211857 A292803 * A292801 A353396 A355533

Adjacent sequences: A348709 A348710 A348711 * A348713 A348714 A348715

Keyword

nonn,tabl,easy

Author

Mohammed Yaseen, Oct 31 2021

Status

approved