A164831 -id:A164831 - cp's OEIS frontend

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

Christopher Hunt Gribble, Aug 27 2009

			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

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

Cf. A164822, A164831, A164846, A165252.

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

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

A165252 Triangle read by rows: Frequency with which the values of cos(2*Pi/k) mod j, listed in A164823, occur for each j >= 1, taken over k = 1..j-1.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 1, 0, 2, 0, 5, 2, 1, 2, 2, 0, 6, 0, 2, 1, 0, 3, 0, 7, 1, 3, 1, 3, 1, 3, 0, 8, 1, 2, 2, 1, 2, 2, 4, 0, 9, 1, 1, 4, 2, 4, 3, 1, 4, 0, 10, 1, 0, 1, 3, 1, 2, 0, 2, 5

Offset: 1

Christopher Hunt Gribble, Sep 10 2009

The FORTRAN program used to generate this sequence is too complex to be listed here.

			The triangle of numbers is:
..j....Frequency of values over k
.......0..1..2..3..4..5..6..7..8..9..10
..1....0
..2....0..1
..3....0..2..1
..4....0..3..0..1
..5....0..4..1..0..2
..6....0..5..2..1..2..2
..7....0..6..0..2..1..0..3
..8....0..7..1..3..1..3..1..3
..9....0..8..1..2..2..1..2..2..4
.10....0..9..1..1..4..2..4..3..1..4
.11....0.10..1..0..1..3..1..2..0..2..5

C. H. Gribble, Flattened triangle, for j = 1..100 and k = 1..j-1.

Cf: A164822, A164823, A164831, A164846.

A164822 Triangle read by rows, giving the number of solutions mod j of T_k(x) = 1, for j >= 2 and k = 1:j-1, where T_k is the k'th Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 4, 1, 5, 1, 1, 2, 2, 2, 1, 4, 1, 4, 1, 7, 1, 4, 1, 1, 2, 3, 4, 1, 6, 1, 4, 1, 4, 2, 5, 1, 8, 1, 5, 2, 1, 2, 2, 2, 3, 4, 1, 2, 2, 6, 1, 4, 1, 11, 1, 4, 1, 11, 1, 4, 1, 1, 2, 2, 2, 1, 4, 4, 2, 2, 2, 1, 6, 1, 4, 2, 5, 1, 8, 1, 9, 2, 4, 1, 9, 1, 1, 4, 2, 8, 1, 8, 1, 8, 2, 4, 1, 14, 1

Offset: 1

Christopher Hunt Gribble, Aug 27 2009

T_k(0) = 1 if k == 0 mod 4, but x=0 is not counted as a solution. - Robert Israel, Apr 06 2015

			The triangle of numbers is:
.....k..1..2..3..4..5..6..7..8..9.10
..j..
..2.....1
..3.....1..2
..4.....1..2..1
..5.....1..2..2..2
..6.....1..4..1..5..1
..7.....1..2..2..2..1..4
..8.....1..4..1..7..1..4..1
..9.....1..2..3..4..1..6..1..4
.10.....1..4..2..5..1..8..1..5..2
.11.....1..2..2..2..3..4..1..2..2..6

C. H. Gribble, Flattened triangle, for j = 2:100 and k = 1:j-1.

Cf. A045468, A164823, A164831, A164846, A165252.

seq(seq(nops(select(t -> orthopoly[T](k, t)-1 mod j = 0, [$1..j-1])), k=1..j-1), j=2..20); # Robert Israel, Apr 06 2015

Table[Length[Select[Range[j-1], Mod[ChebyshevT[k, #]-1, j] == 0&]], {j, 2, 20}, {k, 1, j-1}] // Flatten (* Jean-François Alcover, Mar 27 2019, after Robert Israel *)

From Robert Israel, Apr 06 2015 (Start):
a(k,j) is multiplicative in j for each odd k.
a(k,j)+1 is multiplicative in j for k divisible by 4.
a(k,j)+[j=2] is multiplicative in j for k == 2 mod 4, where [j=2] = 1 if j=2, 0 otherwise.
a(1,j) = 1.
a(2,j) = A060594(j) if j is odd, A060594(j/2) if j is even.
a(3,2^m) = 1.
a(3,p^m) = p^floor(m/2)+1 if p is a prime > 3.
a(4,p^m) = p^floor(m/2)+1 if p is a prime > 2.
a(5,p) = 3 if p is in A045468, 1 for other primes p. (End)

Sequence and definition corrected by Christopher Hunt Gribble, Sep 10 2009

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

A164846 Triangle read by rows: frequency with which the number of values taken by cos(2*Pi/k) mod j occurs for each j >= 1, taken over k = 1..j-1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 1, 3, 0, 0, 0, 3, 0, 0, 1, 1, 0, 2, 3, 0, 1, 0, 0, 0, 4, 0, 0, 2, 0, 0, 1, 0, 3, 1, 1, 2, 0, 1, 0, 0, 0, 3, 2, 0, 1, 2, 0, 0, 1, 0, 0, 2, 5, 1, 1, 0, 1, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 3, 6, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 2, 0, 2, 1, 0, 0, 1, 2, 0, 0, 0, 0

Offset: 1

Christopher Hunt Gribble, Aug 28 2009

See A164823 for explanatory comment and the definition of "cos(x) mod j".

			The triangle of numbers is:
..j....Frequency of numbers of solutions over k
.......0..1..2..3..4..5..6..7..8..9..10
..1....0
..2....0..1
..3....0..1..1
..4....0..2..1..0
..5....0..1..3..0..0
..6....0..3..0..0..1..1
..7....0..2..3..0..1..0..0
..8....0..4..0..0..2..0..0..1
..9....0..3..1..1..2..0..1..0..0
.10....0..3..2..0..1..2..0..0..1..0
.11....0..2..5..1..1..0..1..0..0..0..0

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

Cf. A164822, A164823, A164831, A165252.

Minor edits from N. J. A. Sloane, Sep 01 2009
Sequence corrected by Christopher Hunt Gribble, Sep 10 2009
Minor edits by Christopher Hunt Gribble, Oct 01 2009

cp's OEIS Frontend

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.

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A165252 Triangle read by rows: Frequency with which the values of cos(2*Pi/k) mod j, listed in A164823, occur for each j >= 1, taken over k = 1..j-1.

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A164822 Triangle read by rows, giving the number of solutions mod j of T_k(x) = 1, for j >= 2 and k = 1:j-1, where T_k is the k'th Chebyshev polynomial of the first kind.

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A164846 Triangle read by rows: frequency with which the number of values taken by cos(2*Pi/k) mod j occurs for each j >= 1, taken over k = 1..j-1.

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