A064520 a(n) = + 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 - 10 + 11 + 12 + 13 + 14 + 15 - ... + (+-1)*n, where there is one plus, two minuses, three pluses, etc. (see A002024).

1, -1, -4, 0, 5, 11, 4, -4, -13, -23, -12, 0, 13, 27, 42, 26, 9, -9, -28, -48, -69, -47, -24, 0, 25, 51, 78, 106, 77, 47, 16, -16, -49, -83, -118, -154, -117, -79, -40, 0, 41, 83, 126, 170, 215, 169, 122, 74, 25, -25, -76, -128, -181, -235, -290, -234, -177, -119, -60, 0, 61, 123, 186, 250, 315, 381, 314, 246, 177

Offset: 1

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 07 2001

Keywords

• sign
• look
• easy

Comments

|a(n)| takes its locally maximal values when n is a triangular number, the maximal values being given by A019298.

The maximal positive/negative values occur for n = 1, 3, 6, 10, 15, 21 ... the triangular numbers and are a(n) = 1, -4, 11, -23, 42, -69,106, 215, 381, 616 ... +- int(sqrt(n^3/2) + 0.22098 * sqrt(n)). a(n) = n for n = 5, 13, 25, 41, 61, 85, ... m*(m*2-2)+1 and the previous number is equal to 0. Positive numbers which do not occur in this sequence are 2, 3, 6, 7, 8, 10, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, ...

Examples

			a(9) = -13 because 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 = -13.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002024, A019298, A064528.

Programs

Maple

a := proc(n) option remember: if n=1 then RETURN(1) fi: a(n-1) + n*(-1)^( floor(1/2 + sqrt(2*n)+1)); end: for n from 1 to 150 do printf(`%d,`,a(n)) od:

Mathematica

Accumulate[Flatten[Table[(-1)^(n+1) Range[(n(n-1))/2+1,(n(n+1))/2], {n,15}]]] (* Harvey P. Dale, Apr 22 2015 *)

PARI

t(n) = floor(1/2+sqrt(2*n))
for(n=1,200,print1(sum(k=1,n,(-1)^(t(k)+1)*k)," "))

PARI

t(n)= { floor(sqrt(2*n) + 1/2) }
{ for (n=1, 1000, a=sum(k=1, n, (-1)^(t(k) + 1)*k); write("b064520.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 17 2009

Python

from math import isqrt
def A064520(n): return sum(k if (isqrt(k<<3)+1>>1)&1 else -k for k in range(1,n+1)) # Chai Wah Wu, Oct 16 2022

Formula

a(n) = Sum_{k=1..n} (-1)^(A002024(k)+1)*k.

Extensions

More terms from James Sellers, Jason Earls and Vladeta Jovovic, Oct 08 2001