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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. 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS C. H. 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(2pi) as a polynomial P in cos(2pi/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: A051340 A216764 A165430 * 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

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Last modified January 23 04:16 EST 2020. Contains 331168 sequences. (Running on oeis4.)