OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..10000
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.
FORMULA
Multiplicative with a(2^e) = 2^e for e <= 2, a(2^e) = 2^(e + 1) for e > 2, a(p^e) = (p-1)*p^(e-1) for p-2 mod 8 = +-1, a(p^e) = (p+1)*p^(e-1) for p-2 mod 8 = +-3. - Andrew Howroyd, Jul 13 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = (9/(16*A309710)) = 0.528300880442971272... . - Amiram Eldar, Nov 21 2023
MAPLE
A088965 := proc(n) local a, x, y ; a := 0 ; for x from 0 to n-1 do for y from 0 to n-1 do if (x^2+2*y^2) mod n = 1 mod n then a := a+1 ; end if; end do; end do ; a ; end proc:
seq(A088965(n), n=1..70) ; # R. J. Mathar, Jan 07 2011
MATHEMATICA
a[1]=1; a[n_]:=Length@Rest@Union@Flatten@Table[If[Mod[i^2 + 2 j^2, n]==1, i+I j, 0], {i, 0, n-1}, {j, 0, n-1}]; Table[a[n], {n, 1, 80}] (* Vincenzo Librandi, Jul 16 2018 *)
PROG
(PARI) a(n)={my(v=vector(n)); for(i=0, n-1, v[i^2%n + 1]++); sum(i=0, n-1, v[i+1]*v[(1-2*i)%n + 1])} \\ Andrew Howroyd, Jul 09 2018
(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 2^e*if(e>2, 2, 1), p^(e-1)*if(abs(p%8-2)==1, p-1, p+1)))} \\ Andrew Howroyd, Jul 09 2018
(Magma) [n eq 1 select 1 else #[x: x in [1..n], y in [1..n] | (x^2+2*y^2) mod n eq 1]: n in [1..80]]; // Vincenzo Librandi, Jul 16 2018
CROSSREFS
KEYWORD
mult,nonn,easy
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 28 2003
STATUS
approved