A264081 The sum of the 2 X 2 idempotent matrices over Z/nZ is congruent to {{a(n),0}, {0,a(n)}} (mod n).
0, 0, 1, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 8, 14, 1, 1, 8, 1, 16, 7, 8, 1, 14, 1, 8, 1, 26, 1, 22, 1, 1, 14, 8, 18, 26, 1, 8, 1, 8, 1, 14, 1, 26, 5, 8, 1
Offset: 1
Examples
The 2 X 2 idempotent matrices over Z/3Z are {{0, 0}, {0, 0}}, {{0, 0}, {0,1}}, {{0, 0}, {1, 1}}, {{0, 0}, {2, 1}}, {{0, 1}, {0, 1}}, {{0, 2}, {0, 1}}, {{1, 0}, {0, 0}}, {{1, 0}, {0, 1}}, {{1, 0}, {1, 0}}, {{1, 0}, {2, 0}}, {{1, 1}, {0, 0}}, {{1, 2}, {0, 0}}, {{2, 1}, {1, 2}}, {{2, 2}, {2, 2}}. Their sum is {{10, 9}, {9, 10}} == 1*{{1, 0}, {0, 1}} (mod 3) and therefore a(3) = 1.
Crossrefs
Cf. A226756.
Programs
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Mathematica
K[n_] := K[n] = Mod[Sum[If[ Mod[{{a, b}, {c, d}}.{{a, b}, {c, d}} - {{a, b}, {c, d}}, n] == 0{{a, b}, {c, d}}, {{a, b}, {c, d}}, 0], {a, n}, {b, n}, {c, n}, {d, n}], n]; Table[K[n][[1, 1]], {n, 1, 22}] -
PARI
a(n) = lift(sum(i=0, n-1, sum(j=0, n-1, sum(k=0, n-1, sum(l=0, n-1, m = Mod([i,j;k,l], n); if ((m^2 == m), m[1,1])))))); \\ Michel Marcus, Apr 04 2016