A270439 Alternating sum of nonsquares (A000037).

2, -1, 4, -2, 5, -3, 7, -4, 8, -5, 9, -6, 11, -7, 12, -8, 13, -9, 14, -10, 16, -11, 17, -12, 18, -13, 19, -14, 20, -15, 22, -16, 23, -17, 24, -18, 25, -19, 26, -20, 27, -21, 29, -22, 30, -23, 31, -24, 32, -25, 33, -26, 34, -27, 35, -28, 37, -29, 38, -30, 39, -31, 40, -32, 41, -33, 42, -34, 43, -35, 44, -36, 46, -37, 47

Offset: 1

Author

Ilya Gutkovskiy, Jul 12 2016

Keywords

• easy
• sign

Comments

Interleaving of nontriangular numbers (A014132) and negative integers (A001478).

Examples

			a(1) = a(2*1-1) = 1 + floor(1/2 + sqrt(2*1)) = 2;
a(2) = a(2*1) = -1;
a(3) = a(2*2-1) = 2 + floor(1/2 + sqrt(2*2)) = 4;
a(4) = a(2*2) = -2;
a(5) = a(2*3-1) = 3 + floor(1/2 + sqrt(2*3)) = 5;
a(6) = a(2*3) = -3, etc.
or
a(1) = 2;
a(2) = 2 - 3 = -1;
a(3) = 2 - 3 + 5 = 4;
a(4) = 2 - 3 + 5 - 6 = -2;
a(5) = 2 - 3 + 5 - 6 + 7 = 5;
a(6) = 2 - 3 + 5 - 6 + 7 - 8 = -3, etc.
(2, 3, 5, 6, 7, 8, ... is the nonsquares).
		

Crossrefs

Cf. A000037, A001478, A014132, A086849.

Programs

Mathematica

Table[Sum[(-1)^(k + 1) (k + Floor[1/2 + Sqrt[k]]), {k, n}], {n, 75}]

PARI

a(n)=if(n%2, sqrtint(4*n-3)+n+2, -n)\2 \\ Charles R Greathouse IV, Aug 03 2016

Python

from math import isqrt
def A270439(n): return (n>>1)+1+(m:=isqrt(n+1))+int(n-m*(m+1)>=0) if n&1 else -(n>>1) # Chai Wah Wu, Nov 14 2022

Formula

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

a(n) = Sum_{k = 1..n} (-1)^(k+1)*A000037(k).

a(2^k*m) = -2^(k-1) * m, k > 0.

a(2k - 1) = k + floor(1/2 + sqrt(2*k)), a(2k) = -k, k > 0.

a(2k - 1) = A014132(k), a(2k) = A001478(k).