Search: a032514 -id:a032514
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A268173
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a(n) = Sum_{k=0..n} (-1)^k*floor(sqrt(k)).
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+10
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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OFFSET
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0,10
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LINKS
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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
a(n) = (-1)^n*floor((sqrt(n) + (n mod 2))/2);
a(2*n) = floor(sqrt(n/2));
a(2*n+1) = -floor(sqrt((n+1)/2) + 1/2). (End)
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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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MAPLE
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seq(add((-1)^k*floor(sqrt(k)), k=0..n), n=0..80); # Ridouane Oudra, Jan 21 2024
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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.
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A270370
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a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/3)).
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+10
2
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0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 2, -2, 2, -2, 2, -2
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OFFSET
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0,28
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LINKS
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FORMULA
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a(n) = floor(n^(1/3))*(-1)^n/2 - ((-1)^(floor(n^(1/3))+1)+1)/4.
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EXAMPLE
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a(5) = [0^(1/3)]-[1^(1/3)]+[2^(1/3)]-[3^(1/3)]+[4^(1/3)]-[5^(1/3)] = 0-1+1-1+1-1 = -1, letting [] denote the floor function.
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MATHEMATICA
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Print[Table[Sum[(-1)^i*Floor[i^(1/3)], {i, 0, n}], {n, 0, 100}]]
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PROG
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(PARI) a(n)=sum(i=0, n, (-1)^i*sqrtnint(i, 3))
(PARI) a(n)=sqrtnint(n, 3)*(-1)^n/2-((-1)^(sqrtnint(n, 3)+1)+1)/4
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CROSSREFS
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Cf. A268173, A022554, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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A262352
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a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/4)).
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+10
1
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0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1
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OFFSET
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0,82
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LINKS
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FORMULA
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a(n) = floor(n^(1/4))*(-1)^n/2-((-1)^(floor(n^(1/4))+1)+1)/4.
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EXAMPLE
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Letting [] denote the floor function, a(7) = [0^(1/4)] - [1^(1/4)] + [2^(1/4)] - [3^(1/4)] + [4^(1/4)] - [5^(1/4)] + [6^(1/4)] - [7^(1/4)] = 0 - 1 + 1 - 1 + 1 - 1 + 1 - 1 = -1.
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MATHEMATICA
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Print[Table[Sum[(-1)^k*Floor[k^(1/4)], {k, 0, n}], {n, 0, 100}]] ;
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PROG
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(PARI) a(n)=floor(n^(1/4))*(-1)^n/2-((-1)^(floor(n^(1/4))+1)+1)/4
(PARI) a(n)=sum(k=0, n, (-1)^k*floor(k^(1/4)))
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CROSSREFS
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Cf. A270370, A268173, A022554, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A270825
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a(n) = Sum_{i=0..n} (-1)^floor(i/2)*floor(sqrt(i)).
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+10
0
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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, -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, 3, 10, 3, -4, 4, 12, 4, -4, 4, 12, 4, -4, 4, 12, 4, -4, 4
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,6
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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MATHEMATICA
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Print[Table[Sum[(-1)^(Floor[i/2])*Floor[Sqrt[i]], {i, 0, n}], {n, 0, 100}]]
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PROG
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(PARI) a(n)=sum(i=0, n, (-1)^(floor(i/2))*floor(sqrt(i)))
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CROSSREFS
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Cf. A268173, A022554, A270370, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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