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.
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
Examples
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
Links
- C. H. Gribble, Flattened triangle, for j = 2:100 and k = 1:j-1.
Programs
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Maple
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
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Mathematica
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 *)
Formula
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(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)
Extensions
Sequence and definition corrected by Christopher Hunt Gribble, Sep 10 2009
Minor edit by N. J. A. Sloane, Sep 13 2009
Comments