OFFSET
1,3
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.
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).
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)
EXAMPLE
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.
MATHEMATICA
LinearRecurrence[{0, 2, 0, -1}, {1, -1, 4, -2}, 60] (* Metin Sariyar, Sep 15 2019 *)
PROG
(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)
(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
(Magma) [((2*n-1)*(n mod 2) - n*(-1)^n)/2: n in [0..70]]; // G. C. Greubel, Mar 14 2024
(SageMath) [((2*n-1)*(n%2) - n*(-1)^n)/2 for n in range(1, 71)] # G. C. Greubel, Mar 14 2024
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Mark Povich, Aug 27 2019
STATUS
approved