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A385104 Triangle read by rows: T(n,k) is the number of residue classes obtained by solving mod(x^2,n) = k for x over the integers, n >= 1, k >= 0.

%I A385104 #13 Jun 25 2025 10:19:43
%S A385104 1,1,1,1,2,0,2,2,0,0,1,2,0,0,2,1,2,0,1,2,0,1,2,2,0,2,0,0,2,4,0,0,2,0,
%T A385104 0,0,3,2,0,0,2,0,0,2,0,1,2,0,0,2,1,2,0,0,2,1,2,0,2,2,2,0,0,0,2,0,2,4,
%U A385104 0,0,4,0,0,0,0,2,0,0,1,2,0,2,2,0,0,0,0,2,2,0,2,1,2,2,0,2,0,0,1,2
%N A385104 Triangle read by rows: T(n,k) is the number of residue classes obtained by solving mod(x^2,n) = k for x over the integers, n >= 1, k >= 0.
%C A385104 The sum of each row is n.
%H A385104 Jason Bard, <a href="/A385104/b385104.txt">Table of n, a(n) for n = 1..11325</a>
%F A385104 T(n,0) = A000188(n).
%F A385104 T(n,1) = A060594(n).
%F A385104 T(n,n-1) = A000089(n).
%e A385104 Triangle starts:
%e A385104   1
%e A385104   1 1
%e A385104   1 2 0
%e A385104   2 2 0 0
%e A385104   1 2 0 0 2
%e A385104   1 2 0 1 2 0
%e A385104   1 2 2 0 2 0 0
%e A385104   2 4 0 0 2 0 0 0
%e A385104   3 2 0 0 2 0 0 2 0
%e A385104   1 2 0 0 2 1 2 0 0 2
%e A385104   ...
%t A385104 dat[n_] := Table[Reduce[Mod[x^2, n] == k, x, Integers], {k, 0, n - 1}]; countConditions[cond_] := Which[cond === False, 0, MatchQ[cond, x \[Element] Integers], 1, True, Length@Cases[cond, Equal[x, _], Infinity]]; counts = Flatten[Table[countConditions /@ dat[n], {n, 1, 20}]]
%o A385104 (PARI) T(n, k) = sum(i=1, n, Mod(i,n)^2 == k);
%o A385104 row(n) = vector(n, i, T(n, i-1)); \\ _Michel Marcus_, Jun 23 2025
%Y A385104 Cf. A000089, A000188, A060594, A096008.
%K A385104 nonn,tabl
%O A385104 1,5
%A A385104 _Jason Bard_, Jun 18 2025