|
%I #28 Mar 27 2019 10:51:51
%S 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,
%T 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,
%U 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
%N 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.
%C T_k(0) = 1 if k == 0 mod 4, but x=0 is not counted as a solution. - _Robert Israel_, Apr 06 2015
%H C. H. Gribble, <a href="/A164822/b164822.txt">Flattened triangle, for j = 2:100 and k = 1:j-1</a>.
%F From _Robert Israel_, Apr 06 2015 (Start):
%F a(k,j) is multiplicative in j for each odd k.
%F a(k,j)+1 is multiplicative in j for k divisible by 4.
%F a(k,j)+[j=2] is multiplicative in j for k == 2 mod 4, where [j=2] = 1 if j=2, 0 otherwise.
%F a(1,j) = 1.
%F a(2,j) = A060594(j) if j is odd, A060594(j/2) if j is even.
%F a(3,2^m) = 1.
%F a(3,p^m) = p^floor(m/2)+1 if p is a prime > 3.
%F a(4,p^m) = p^floor(m/2)+1 if p is a prime > 2.
%F a(5,p) = 3 if p is in A045468, 1 for other primes p. (End)
%e The triangle of numbers is:
%e .....k..1..2..3..4..5..6..7..8..9.10
%e ..j..
%e ..2.....1
%e ..3.....1..2
%e ..4.....1..2..1
%e ..5.....1..2..2..2
%e ..6.....1..4..1..5..1
%e ..7.....1..2..2..2..1..4
%e ..8.....1..4..1..7..1..4..1
%e ..9.....1..2..3..4..1..6..1..4
%e .10.....1..4..2..5..1..8..1..5..2
%e .11.....1..2..2..2..3..4..1..2..2..6
%p 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
%t 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_ *)
%Y Cf. A045468, A164823, A164831, A164846, A165252.
%K nonn,tabl
%O 1,3
%A _Christopher Hunt Gribble_, Aug 27 2009
%E Sequence and definition corrected by _Christopher Hunt Gribble_, Sep 10 2009
%E Minor edit by _N. J. A. Sloane_, Sep 13 2009
|
