A268173 -id:A268173 - cp's OEIS frontend

A270370 a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/3)).

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

Offset: 0

John M. Campbell, Mar 15 2016

			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.

Cf. A268173, A022554, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.

Print[Table[Sum[(-1)^i*Floor[i^(1/3)],{i,0,n}],{n,0,100}]]

PARI
```
a(n)=sum(i=0,n,(-1)^i*sqrtnint(i,3))
```

a(n)=sqrtnint(n,3)*(-1)^n/2-((-1)^(sqrtnint(n,3)+1)+1)/4

a(n) = floor(n^(1/3))*(-1)^n/2 - ((-1)^(floor(n^(1/3))+1)+1)/4.

A262352 a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/4)).

Original entry on oeis.org

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

Offset: 0

John M. Campbell, Mar 24 2016

			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.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A270370, A268173, A022554, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.

Print[Table[Sum[(-1)^k*Floor[k^(1/4)],{k,0,n}],{n,0,100}]] ;

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)))
```

A262352(n) = sum(k=0,n,((-1)^k)*sqrtnint(k, 4)); \\ Antti Karttunen, Nov 06 2018

a(n) = floor(n^(1/4))*(-1)^n/2-((-1)^(floor(n^(1/4))+1)+1)/4.

More terms from Antti Karttunen, Nov 06 2018

A270825 a(n) = Sum_{i=0..n} (-1)^floor(i/2)*floor(sqrt(i)).

Original entry on oeis.org

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

Offset: 0

John M. Campbell, Mar 23 2016

			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.

Cf. A268173, A022554, A270370, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.

Print[Table[Sum[(-1)^(Floor[i/2])*Floor[Sqrt[i]],{i,0,n}],{n,0,100}]]

a(n)=sum(i=0,n,(-1)^(floor(i/2))*floor(sqrt(i)))

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).

cp's OEIS Frontend

A270370 a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/3)).

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A262352 a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/4)).

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A270825 a(n) = Sum_{i=0..n} (-1)^floor(i/2)*floor(sqrt(i)).

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