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

Author

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

Keywords

• nonn
• mult

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

Crossrefs

Cf. A000224, A068516, A182039, A209411.

Programs

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

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

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