06/11/03 11:02:52
>>13
各線分の長さは1以上異なるから、辺の長さ>1 ∴ 辺の長さ≧2.
階差数列は
b_n = a_n - a_(n-1) = #{(i,j)∈Z^2 | 2≦i<j≦n-1, i+j≧n+1},
b_(2k+1) = k(k-1), b_(2k) = (k-1)^2. (← kについての帰納法で)
nが奇数のとき
a_(2k+1) - a_(2k-1) = b_(2k+1) + b_(2k) = (k-1)(2k-1), a_5=3.
a_n = 3 + ∑[k=3,(n-1)/2] (k-1)(2k-1) = (n-3)(n-1)(2n-1)/24.
nが偶数のとき
a_(2k) - a_(2k-2) = b_(2k) + b_(2k-1) = (k-1)(2k-3), a_4=1.
a_n = 1 + ∑[k=3,n/2] (k-1)(2k-3) = (n-2)n(2n-5)/24.