A164823 Irregular triangle read by rows, listing the values x for which T_k(x) == 1 (mod j) for j >= 2 and k = 1..j-1, where T_k are the Chebyshev polynomials of the first kind.

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 2, 1, 4, 1, 1, 2, 4, 5, 1, 1, 2, 3, 4, 5, 1, 1, 1, 6, 1, 3, 1, 6, 1, 1, 3, 4, 6, 1, 1, 3, 5, 7, 1, 1, 2, 3, 4, 5, 6, 7, 1, 1, 3, 5, 7, 1, 1, 1, 8, 1, 4, 7, 1, 3, 6, 8, 1, 1, 2, 4, 5, 7, 8, 1, 1, 3, 6, 8, 1, 1, 4, 6, 9, 1, 7, 1, 4, 5, 6, 9, 1, 1, 2, 3, 4, 6, 7, 8, 9, 1, 1, 4, 5

Offset

1,4

Links

Christopher Hunt Gribble, Flattened irregular triangle, for j = 2..100 and k = 1..j-1.

Example

The values are listed horizontally in increasing order for each (j, k) under the column headed "cos(2*Pi/k) mod j".
The column headed "nov" is the number of values. The values read downwards form A164822.
I call "cos(x) mod j" the "Discrete Cosine of x modulo j".
cos(2*Pi/k) mod j can be calculated by expressing cos(2*Pi) as a polynomial P in cos(2*Pi/k), for which the coefficients are those of Chebyshev's T(n,x) polynomials (A053120), and then solving P - 1 == 0 (mod j) by trial and error.
...j.......k.....nov....cos(2*Pi/k).mod.j
...2.......1.......1.......1
...3.......1.......1.......1
...........2.......2.......1.......2
...4.......1.......1.......1
...........2.......2.......1.......3
...........3.......1.......1
...5.......1.......1.......1
...........2.......2.......1.......4
...........3.......2.......1.......2
...........4.......2.......1.......4
...6.......1.......1.......1
...........2.......4.......1.......2.......4.......5
...........3.......1.......1
...........4.......5.......1.......2.......3.......4.......5
...........5.......1.......1
...7.......1.......1.......1
...........2.......2.......1.......6
...........3.......2.......1.......3
...........4.......2.......1.......6
...........5.......1.......1
...........6.......4.......1.......3.......4.......6
...8.......1.......1.......1
...........2.......4.......1.......3.......5.......7
...........3.......1.......1
...........4.......7.......1.......2.......3.......4.......5.......6.......7
...........5.......1.......1
...........6.......4.......1.......3.......5.......7
...........7.......1.......1
...9.......1.......1.......1
...........2.......2.......1.......8
...........3.......3.......1.......4.......7
...........4.......4.......1.......3.......6.......8
...........5.......1.......1
...........6.......6.......1.......2.......4.......5.......7.......8
...........7.......1.......1
...........8.......4.......1.......3.......6.......8
..10.......1.......1.......1
...........2.......4.......1.......4.......6.......9
...........3.......2.......1.......7
...........4.......5.......1.......4.......5.......6.......9
...........5.......1.......1
...........6.......8.......1.......2.......3.......4.......6.......7.......8.......9
...........7.......1.......1
...........8.......5.......1.......4.......5.......6.......9
...........9.......2.......1.......7
..11.......1.......1.......1
...........2.......2.......1......10
...........3.......2.......1.......5
...........4.......2.......1......10
...........5.......3.......1.......7.......9
...........6.......4.......1.......5.......6......10
...........7.......1.......1
...........8.......2.......1......10
...........9.......2.......1.......5
..........10.......6.......1.......2.......4.......7.......9......10

Maple

seq(seq(seq(`if`(orthopoly[T](k, t)-1 mod j = 0, t, NULL), t=1..j-1), k=1..j-1), j=2..20); # Robert Israel, Apr 06 2015

Crossrefs

Cf. A164822, A164831, A164846, A165252.

Sequence in context: A216764 A165430 A334215 * A167269 A105535 A182980

Adjacent sequences: A164820 A164821 A164822 * A164824 A164825 A164826

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Aug 27 2009

Extensions

Sequence corrected by Christopher Hunt Gribble, Sep 10 2009

Minor edit by N. J. A. Sloane, Sep 13 2009

Minor edit by Christopher Hunt Gribble, Oct 01 2009

Status

approved