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%I #21 Apr 16 2014 05:39:02
%S 1,4,9,31,88,288,889,2884,9211,29976,97296,318371,1042756,3429604,
%T 11298969,37320679,123473176
%N Number of sequences of length n whose terms are positive integers less than or equal to n in which the i-th term is greater than both the (i-2)nd and (i-3)rd terms.
%C Apparently a(n) = sum(k=0..n, C(k, n-k)*C(2*n-k, n) ). - _Paul Barry_, Dec 02 2008
%D Romanian Informatics Olympiad, 2001
%e The valid sequences for n = 3 are (1, 1, 2), (1, 1, 3), (1, 2, 2), (1, 2, 3), (1, 3, 2), (1, 3, 3), (2, 1, 3), (2, 2, 3), (2, 3, 3).
%o (C)
%o #include <stdio.h>
%o #define MAX_N 1001
%o int N;
%o int DP[MAX_N][MAX_N], X[MAX_N][MAX_N];
%o int main() {
%o int i, j;
%o scanf("%d ", &N);
%o for(i = 1; i <= N; i++) { DP[1][i] = i; DP[2][i] = i * i; }
%o for(i = 3; i <= N; i++) {
%o for(j = 1; j <= N; j++) {
%o if(j - 2 >= 0) X[i][j] = X[i][j - 1] + DP[i - 2][j - 2];
%o DP[i][j] = DP[i][j - 1] + DP[i - 1][j - 1]
%o + DP[i - 2][j - 1] + X[i][j];
%o }
%o }
%o printf("%d ", DP[N][N]);
%o return 0;
%o }
%K nonn,more
%O 1,2
%A _Shravas Rao_, Jul 30 2008
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