cp's OEIS Frontend

A227553 Number of solutions to x^2 - y^2 - z^2 == 1 (mod n).

%I A227553 #22 Jun 24 2018 11:50:59
%S A227553 1,4,6,8,30,24,42,32,54,120,110,48,182,168,180,128,306,216,342,240,
%T A227553 252,440,506,192,750,728,486,336,870,720,930,512,660,1224,1260,432,
%U A227553 1406,1368,1092,960,1722,1008,1806,880,1620,2024,2162,768,2058,3000,1836
%N A227553 Number of solutions to x^2 - y^2 - z^2 == 1 (mod n).
%C A227553 Conjecture: a(2) = 4; if s > 1 then a(2^s) = 2^(2s-1); if p == 1 (mod 4) then a(p^s) = (p+1)*p^(2s-1); if p == 3 (mod 4) then a(p^s) = (p-1)*p^(2s-1).
%H A227553 Andrew Howroyd, <a href="/A227553/b227553.txt">Table of n, a(n) for n = 1..2500</a>
%H A227553 L. Toth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Toth/toth12.html">Counting Solutions of Quadratic Congruences in Several Variables Revisited</a>, J. Int. Seq. 17 (2014) # 14.11.6.
%t A227553 a[1] = 1; a[n_] := Sum[If[Mod[a^2-b^2-c^2, n] == 1, 1, 0], {a, n}, {b, n}, {c, n}]; Table[a[n], {n, 10}]
%o A227553 (PARI)
%o A227553 M(n,f)={sum(i=0, n-1, Mod(x^(f(i)%n), x^n-1))}
%o A227553 a(n)={polcoeff(lift(M(n, i->i^2) * M(n, i->-(i^2))^2 ), 1%n)} \\ _Andrew Howroyd_, Jun 24 2018
%Y A227553 Cf. A208895, A086932, A089003, A060968, A087784.
%K A227553 nonn,mult
%O A227553 1,2
%A A227553 _José María Grau Ribas_, Jul 16 2013