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A268173
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a(n) = Sum_{k=0..n} (-1)^k*floor(sqrt(k)).
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4
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0, -1, 0, -1, 1, -1, 1, -1, 1, -2, 1, -2, 1, -2, 1, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -3, 2, -3, 2, -3, 2, -3, 2, -3, 2, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -4, 3, -4, 3, -4, 3, -4, 3, -4, 3, -4, 3, -4, 3, -4, 4, -4, 4, -4, 4, -4, 4, -4, 4, -4, 4, -4, 4, -4, 4, -4, 4
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refs;
listen;
history;
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OFFSET
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0,10
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) = floor(sqrt(n))*(-1)^n/2 - ((-1)^(floor(sqrt(n))+1)+1)/4.
a(n) = (-1)^n * Sum_{i=1..ceiling(n/2)} c(n+2-2*i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020
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EXAMPLE
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a(5) = -1 = floor(sqrt(0)) - floor(sqrt(1)) + floor(sqrt(2)) - floor(sqrt(3)) + floor(sqrt(4)) - floor(sqrt(5)).
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MATHEMATICA
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Table[Sum[(-1)^k Floor[Sqrt@ k], {k, 0, n}], {n, 0, 50}] (* Michael De Vlieger, Mar 15 2016 *)
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PROG
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(PARI) a(n) = sum(k=0, n, (-1)^k*sqrtint(k)); \\ Michel Marcus, Jan 28 2016
(PARI) a(n) = sqrtint(n)*(-1)^n/2-((-1)^(sqrtint(n)+1)+1)/4; \\ John M. Campbell, Mar 15 2016
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CROSSREFS
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Cf. A022554, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.
Sequence in context: A033105 A106703 A127267 * A008617 A339369 A025824
Adjacent sequences: A268170 A268171 A268172 * A268174 A268175 A268176
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KEYWORD
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sign,easy
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AUTHOR
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John M. Campbell, Jan 28 2016
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EXTENSIONS
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Terms a(55) and beyond from Andrew Howroyd, Mar 02 2020
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STATUS
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approved
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