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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.

%I A164823 #15 Dec 27 2022 02:37:25
%S A164823 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,
%T A164823 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,
%U A164823 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
%N 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.
%H A164823 Christopher Hunt Gribble, <a href="/A164823/b164823.txt">Flattened irregular triangle, for j = 2..100 and k = 1..j-1</a>.
%e A164823 The values are listed horizontally in increasing order for each (j, k) under the column headed "cos(2*Pi/k) mod j".
%e A164823 The column headed "nov" is the number of values. The values read downwards form A164822.
%e A164823 I call "cos(x) mod j" the "Discrete Cosine of x modulo j".
%e A164823 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.
%e A164823 ...j.......k.....nov....cos(2*Pi/k).mod.j
%e A164823 ...2.......1.......1.......1
%e A164823 ...3.......1.......1.......1
%e A164823 ...........2.......2.......1.......2
%e A164823 ...4.......1.......1.......1
%e A164823 ...........2.......2.......1.......3
%e A164823 ...........3.......1.......1
%e A164823 ...5.......1.......1.......1
%e A164823 ...........2.......2.......1.......4
%e A164823 ...........3.......2.......1.......2
%e A164823 ...........4.......2.......1.......4
%e A164823 ...6.......1.......1.......1
%e A164823 ...........2.......4.......1.......2.......4.......5
%e A164823 ...........3.......1.......1
%e A164823 ...........4.......5.......1.......2.......3.......4.......5
%e A164823 ...........5.......1.......1
%e A164823 ...7.......1.......1.......1
%e A164823 ...........2.......2.......1.......6
%e A164823 ...........3.......2.......1.......3
%e A164823 ...........4.......2.......1.......6
%e A164823 ...........5.......1.......1
%e A164823 ...........6.......4.......1.......3.......4.......6
%e A164823 ...8.......1.......1.......1
%e A164823 ...........2.......4.......1.......3.......5.......7
%e A164823 ...........3.......1.......1
%e A164823 ...........4.......7.......1.......2.......3.......4.......5.......6.......7
%e A164823 ...........5.......1.......1
%e A164823 ...........6.......4.......1.......3.......5.......7
%e A164823 ...........7.......1.......1
%e A164823 ...9.......1.......1.......1
%e A164823 ...........2.......2.......1.......8
%e A164823 ...........3.......3.......1.......4.......7
%e A164823 ...........4.......4.......1.......3.......6.......8
%e A164823 ...........5.......1.......1
%e A164823 ...........6.......6.......1.......2.......4.......5.......7.......8
%e A164823 ...........7.......1.......1
%e A164823 ...........8.......4.......1.......3.......6.......8
%e A164823 ..10.......1.......1.......1
%e A164823 ...........2.......4.......1.......4.......6.......9
%e A164823 ...........3.......2.......1.......7
%e A164823 ...........4.......5.......1.......4.......5.......6.......9
%e A164823 ...........5.......1.......1
%e A164823 ...........6.......8.......1.......2.......3.......4.......6.......7.......8.......9
%e A164823 ...........7.......1.......1
%e A164823 ...........8.......5.......1.......4.......5.......6.......9
%e A164823 ...........9.......2.......1.......7
%e A164823 ..11.......1.......1.......1
%e A164823 ...........2.......2.......1......10
%e A164823 ...........3.......2.......1.......5
%e A164823 ...........4.......2.......1......10
%e A164823 ...........5.......3.......1.......7.......9
%e A164823 ...........6.......4.......1.......5.......6......10
%e A164823 ...........7.......1.......1
%e A164823 ...........8.......2.......1......10
%e A164823 ...........9.......2.......1.......5
%e A164823 ..........10.......6.......1.......2.......4.......7.......9......10
%p A164823 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
%Y A164823 Cf. A164822, A164831, A164846, A165252.
%K A164823 nonn,tabf
%O A164823 1,4
%A A164823 _Christopher Hunt Gribble_, Aug 27 2009
%E A164823 Sequence corrected by _Christopher Hunt Gribble_, Sep 10 2009
%E A164823 Minor edit by _N. J. A. Sloane_, Sep 13 2009
%E A164823 Minor edit by _Christopher Hunt Gribble_, Oct 01 2009