A268173 -id:A268173

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A270370

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,28

LINKS

Table of n, a(n) for n=0..69.

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(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

CROSSREFS

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

KEYWORD

sign,easy

AUTHOR

John M. Campbell, Mar 15 2016

STATUS

approved

A262352

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,82

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16383

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

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(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)))

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

CROSSREFS

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

KEYWORD

sign,easy

AUTHOR

John M. Campbell, Mar 24 2016

EXTENSIONS

More terms from Antti Karttunen, Nov 06 2018

STATUS

approved

A270825

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

+10
0

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)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..76.

FORMULA

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).

EXAMPLE

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.

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

sign,easy

AUTHOR

John M. Campbell, Mar 23 2016

STATUS

approved

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