A255563 a(n) = -3 * n/4 if n divisible by 4, a(n) = -(-1)^n * n otherwise.

0, 1, -2, 3, -3, 5, -6, 7, -6, 9, -10, 11, -9, 13, -14, 15, -12, 17, -18, 19, -15, 21, -22, 23, -18, 25, -26, 27, -21, 29, -30, 31, -24, 33, -34, 35, -27, 37, -38, 39, -30, 41, -42, 43, -33, 45, -46, 47, -36, 49, -50, 51, -39, 53, -54, 55, -42, 57, -58, 59

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Michael Somos, Rational Function Multiplicative Coefficients.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Formula

Euler transform of length 10 sequence [-2, 2, 1, 2, 1, -1, 0, 0, 0, -1].

a(n) is multiplicative with a(2) = -2, a(2^e) = -3*2^(e-2) if e>1, a(p^e) = p^e otherwise.

For nonnegative n, |a(n)| counts the distinct differences of perfect squares (mod n).

G.f.: -f(-x) + f(x^4) where f(x) := x / (1 - x)^2.

G.f.: x * (1 + x^3) * (1 + x^5) / ((1 + x) * (1 - x^4))^2.

a(n) = -a(-n) for all n in Z.

From Amiram Eldar, Dec 29 2022: (Start)

Dirichlet g.f.: zeta(s-1)*(1-2^(2-s)+4^(-s)).

Sum_{k=1..n} a(k) ~ n^2/32. (End)

Example

G.f. = x - 2*x^2 + 3*x^3 - 3*x^4 + 5*x^5 - 6*x^6 + 7*x^7 - 6*x^8 + 9*x^9 + ...

Mathematica

a[ n_] := n {1, -1, 1, -3/4}[[Mod[ n, 4, 1]]];
a[ n_] := n If[ Divisible[ n, 4], -3/4, -(-1)^n];
CoefficientList[Series[x*(1+x^3)*(1+x^5)/((1+x)*(1-x^4))^2, {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)

Prog

(PARI) {a(n) = n * [-3/4, 1, -1, 1][n%4 + 1]};
(PARI) {a(n) = n * if( n%4, -(-1)^n, -3/4)};
(PARI) my(x='x+O('x^60)); Vec(x*(1+x^3)*(1+x^5)/((1+x)*(1-x^4))^2) \\ G. C. Greubel, Aug 02 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x * (1+x^3)*(1+x^5)/((1+x)*(1-x^4))^2)); // G. C. Greubel, Aug 02 2018

Crossrefs

Sequence in context: A359588 A337868 A063659 * A331288 A115350 A320034

Adjacent sequences: A255560 A255561 A255562 * A255564 A255565 A255566

Keyword

sign,mult,easy

Author

Michael Somos, May 05 2015

Status

approved