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

1, 4, 6, 8, 30, 24, 42, 32, 54, 120, 110, 48, 182, 168, 180, 128, 306, 216, 342, 240, 252, 440, 506, 192, 750, 728, 486, 336, 870, 720, 930, 512, 660, 1224, 1260, 432, 1406, 1368, 1092, 960, 1722, 1008, 1806, 880, 1620, 2024, 2162, 768, 2058, 3000, 1836

Offset

1,2

Comments

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).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..2500

L. Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014) # 14.11.6.

Mathematica

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}]

Prog

(PARI)
M(n, f)={sum(i=0, n-1, Mod(x^(f(i)%n), x^n-1))}
a(n)={polcoeff(lift(M(n, i->i^2) * M(n, i->-(i^2))^2 ), 1%n)} \\ Andrew Howroyd, Jun 24 2018

Crossrefs

Cf. A208895, A086932, A089003, A060968, A087784.

Sequence in context: A300658 A366557 A089330 * A108270 A019161 A291718

Adjacent sequences: A227550 A227551 A227552 * A227554 A227555 A227556

Keyword

nonn,mult

Author

José María Grau Ribas, Jul 16 2013

Status

approved