A068197 Number of squares (of another matrix) in M_2(n) - the ring of 2 X 2 matrices over Z_n.

1, 10, 29, 48, 223, 290, 865, 344, 1587, 2230, 5341, 1392, 10459, 8650, 6467, 3182, 30745, 15870, 48061, 10704, 25085, 53410, 103489, 9976, 108035, 104590, 118179, 41520, 262291, 64670, 342721, 41736, 154889, 307450, 192895, 76176, 696655, 480610, 303311, 76712, 1051261, 250850, 1272349, 256368, 353901

Offset

1,2

Comments

a(n) is multiplicative. This is the 2-dimensional analog of A000224.

Links

Giovanni Resta, Table of n, a(n) for n = 1..210

Prog

(Sage)
def A68197(n):
    S = set()
    L = list(range(n))
    for a, b, c, d in cartesian_product([L, L, L, L]):
        M = Matrix([[a, b], [c, d]])
        N = tuple(x % n for x in (M * M).list())
        if N not in S:
           S.add(N)
    print(n, len(S)) # Manfred Scheucher, Jun 12 2015
(PARI) a(n)={my(M=Map()); for(a=0, n-1, for(b=0, n-1, for(c=0, n-1, for(d=0, n-1, mapput(M, lift(Mod([a, b; c, d], n)^2), 1))))); #M} \\ Andrew Howroyd, Aug 06 2018

Crossrefs

Cf. A000224, A068516, A182039, A209411.

Sequence in context: A098751 A143190 A009771 * A079273 A271991 A048469

Adjacent sequences: A068194 A068195 A068196 * A068198 A068199 A068200

Keyword

nonn,mult

Author

Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002

Extensions

More terms from Manfred Scheucher, Jun 12 2015

a(45) corrected by Giovanni Resta, Jun 12 2015

a(1) added by Andrew Howroyd, Aug 06 2018

Status

approved