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%I #8 Apr 05 2016 00:33:35
%S 0,1,0,-1,1,3,1,-1,1,4,1,-2,1,4,1,-2,2,6,2,-2,2,6,2,-2,2,7,2,-3,2,7,2,
%T -3,2,7,2,-3,3,9,3,-3,3,9,3,-3,3,9,3,-3,3,10,3,-4,3,10,3,-4,3,10,3,-4,
%U 3,10,3,-4,4,12,4,-4,4,12,4,-4,4,12,4,-4,4
%N a(n) = Sum_{i=0..n} (-1)^floor(i/2)*floor(sqrt(i)).
%F a(4m)=floor(sqrt(m)), a(4m+1)=floor(3/2*floor(sqrt(4m+1))), a(4m+2)=floor(sqrt(m)), a(4m+3)=-floor((1+sqrt(4m+3))/2).
%e Letting [] denote the floor function, a(7) = [sqrt(0)]+[sqrt(1)]-[sqrt(2)]-[sqrt(3)]+[sqrt(4)]+[sqrt(5)]-[sqrt(6)]-[sqrt(7)] = 0+1-1-1+2+2-2-2 = -1.
%t Print[Table[Sum[(-1)^(Floor[i/2])*Floor[Sqrt[i]],{i,0,n}],{n,0,100}]]
%o (PARI) a(n)=sum(i=0,n,(-1)^(floor(i/2))*floor(sqrt(i)))
%Y Cf. A268173, A022554, A270370, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.
%K sign,easy
%O 0,6
%A _John M. Campbell_, Mar 23 2016
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