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


%S 1,10,29,48,223,290,865,344,1587,2230,5341,1392,10459,8650,6467,3182,

%T 30745,15870,48061,10704,25085,53410,103489,9976,108035,104590,118179,

%U 41520,262291,64670,342721,41736,154889,307450,192895,76176,696655,480610,303311,76712,1051261,250850,1272349,256368,353901

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

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

%H Giovanni Resta, <a href="/A068197/b068197.txt">Table of n, a(n) for n = 1..210</a>

%o (Sage)

%o def A68197(n):

%o S = set()

%o L = list(range(n))

%o for a, b, c, d in cartesian_product([L, L, L, L]):

%o M = Matrix([[a, b], [c, d]])

%o N = tuple(x % n for x in (M * M).list())

%o if N not in S:

%o S.add(N)

%o print(n, len(S)) # _Manfred Scheucher_, Jun 12 2015

%o (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

%Y Cf. A000224, A068516, A182039, A209411.

%K nonn,mult

%O 1,2

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

%E More terms from _Manfred Scheucher_, Jun 12 2015

%E a(45) corrected by _Giovanni Resta_, Jun 12 2015

%E a(1) added by _Andrew Howroyd_, Aug 06 2018

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Last modified July 12 20:45 EDT 2020. Contains 335669 sequences. (Running on oeis4.)