cp's OEIS Frontend

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

%I A068197 #22 Mar 20 2020 19:21:20
%S A068197 1,10,29,48,223,290,865,344,1587,2230,5341,1392,10459,8650,6467,3182,
%T A068197 30745,15870,48061,10704,25085,53410,103489,9976,108035,104590,118179,
%U A068197 41520,262291,64670,342721,41736,154889,307450,192895,76176,696655,480610,303311,76712,1051261,250850,1272349,256368,353901
%N A068197 Number of squares (of another matrix) in M_2(n) - the ring of 2 X 2 matrices over Z_n.
%C A068197 a(n) is multiplicative. This is the 2-dimensional analog of A000224.
%H A068197 Giovanni Resta, <a href="/A068197/b068197.txt">Table of n, a(n) for n = 1..210</a>
%o A068197 (Sage)
%o A068197 def A68197(n):
%o A068197     S = set()
%o A068197     L = list(range(n))
%o A068197     for a, b, c, d in cartesian_product([L, L, L, L]):
%o A068197         M = Matrix([[a, b], [c, d]])
%o A068197         N = tuple(x % n for x in (M * M).list())
%o A068197         if N not in S:
%o A068197            S.add(N)
%o A068197     print(n, len(S)) # _Manfred Scheucher_, Jun 12 2015
%o A068197 (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 A068197 Cf. A000224, A068516, A182039, A209411.
%K A068197 nonn,mult
%O A068197 1,2
%A A068197 Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002
%E A068197 More terms from _Manfred Scheucher_, Jun 12 2015
%E A068197 a(45) corrected by _Giovanni Resta_, Jun 12 2015
%E A068197 a(1) added by _Andrew Howroyd_, Aug 06 2018