mを4で割った余りで場合分けする。
m = 4kのとき
n = 12k, a = 6k, S1 = 9k2
b = 2k-1, S2 = 3k2, S3 = 6k-1
m = 4k+1のとき
n = 12k+3, a = 6k+1, S1 = 9k2+3k
b = 2k, S2 = 3k2+3k+1, S3 = 6k+1
m = 4k+2のとき
n = 12k+6, a = 6k+3, S1 = 9k2+9k+2
b = 2k, S2 = 3k2+3k+1, S3 = 6k+2
m = 4k+3のとき
n = 12k+9, a = 6k+4, S1 = 9k2+12k+4
b = 2k+1, S2 = 3k2+6k+3, S3 = 6k+4
m = 4kのとき
S1+S2 = 3/4m2, S3 = 3/2m-1
m = 4k+1のとき
S1+S2 = 3/4m2+1/4, S3 = 3/2m-1/2
m = 4k+2のとき
S1+S2 = 3/4m2, S3 = 3/2m-1
m = 4k+3のとき
S1+S2 = 3/4m2+1/4, S3 = 3/2m-1/2
いずれの場合も、
S = 2(S1+S2)-S3 = 3/2m2-3/2m+1
となる。
nが3で割り切れないときも同様、のはず。