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In this correspondence, new binary sequence families Fk of period 2n-1 are constructed for even n and any k with gcd(k,n)=2 if n/2 is odd or gcd(k,n)=1 if n/2 is even. The distribution of their correlation values is completely determined. These families have maximum correlation 2n/2+1 and family size 23n/2 + 2n/2 for odd n/2 or 23n/2+2n/2-1 for even n/2. The proposed families include the large set of Kasami sequences, where the k is taken as k=n/2+1.