A319556 a(n) gives the alternating sum of length n, starting at n: n - (n+1) + (n+2) - ... + (-1)^(n+1) * (2n-1).

1, -1, 4, -2, 7, -3, 10, -4, 13, -5, 16, -6, 19, -7, 22, -8, 25, -9, 28, -10, 31, -11, 34, -12, 37, -13, 40, -14, 43, -15, 46, -16, 49, -17, 52, -18, 55, -19, 58, -20, 61, -21, 64, -22, 67, -23, 70, -24, 73, -25, 76, -26, 79, -27, 82, -28, 85, -29, 88, -30

Offset: 1

Author

Mark Povich, Aug 27 2019

Keywords

• sign
• easy

Comments

As can be observed from Bernard Schott's formula, and also proved using elementary methods of slope and angle determination, extending the graph of this sequence forms two lines (given by y = 1.5x - 0.5 and y = -0.5x) that intersect at (0.25, -0.125) in an angle of intersection of ~82.87 degrees. The angles of incidence of these lines off the horizontal axis are ~56.31 and ~-26.56 degrees.

If one wished to include negative input values, one could proceed, e.g., -3+4-5 (=-8) or -3+2-1 (=-2). If the former, then the sequence merely switches signs for negative inputs, graphically extending the previous lines to the left of the vertical. If the latter, two new lines emerge left of the vertical, both of slope 1/2. Increasing the run in this case "spreads apart" all y-intercepts.

Examples

			If n=5, a(n)=7, since 5-6+7-8+9 = 7.
If n=6, a(n)=-3, since 6-7+8-9+10-11 = -3.
		

Links

• Mark Povich, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A094727, A123684, A128622.

Programs

Magma

[((2*n-1)*(n mod 2) - n*(-1)^n)/2: n in [0..70]]; // G. C. Greubel, Mar 14 2024

Mathematica

LinearRecurrence[{0,2,0,-1}, {1,-1,4,-2}, 60] (* Metin Sariyar, Sep 15 2019 *)

PARI

a(n) = sum(k=n, 2*n-1, (-1)^(n-k)*k); \\ Michel Marcus, Aug 27 2019

PARI

Vec(x*(1 - x + 2*x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60)) \\ Colin Barker, Sep 07 2019

Python

def alt(k):
    return sum(k[::2])-sum(k[1::2])
def alt_run(n):
    m = []
    m.append(n)
    for i in range (1, n):
        m.append(m[0]+i)
    return alt(m)
t=[]
for i in range (100):
    t.append(alt_run(i))
print(t)

SageMath

[((2*n-1)*(n%2) - n*(-1)^n)/2 for n in range(1,71)] # G. C. Greubel, Mar 14 2024

Formula

From Bernard Schott, Aug 27 2019: (Start)

a(2*n-1) = 3*n-2 for n >= 1,

a(2*n) = - n for n >= 1. (End)

a(n) = Sum_{k=n..2*n-1} (-1)^(n-k)*k.

From Colin Barker, Sep 07 2019: (Start)

G.f.: x*(1 - x + 2*x^2) / ((1 - x)^2*(1 + x)^2).

a(n) = 2*a(n-2) - a(n-4) for n>4.

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

E.g.f.: (1/4)*((1 + 4*x)*exp(-x) - (1 - 2*x)*exp(x)). - Stefano Spezia, Sep 07 2019 after Colin Barker

From G. C. Greubel, Mar 14 2024: (Start)

a(n) = Sum_{k=0..n-1} (-1)^k*A094727(n, k).

a(n) = Sum_{k=1..n} (-1)^(k-1)*A128622(n, k). (End)