OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).
FORMULA
a(n) is multiplicative with a(2) = 3, a(2^e) = 2^(e-1) if e>1, a(p^e) = p^e if p>2.
Euler transform of length 6 sequence [3, -3, 1, 2, 0, -1].
G.f.: x * (1 + x) * (1 + x^3) / ((1 - x) * (1 + x^2))^2.
G.f.: x * (1 - x^2)^3 * (1 - x^6) / ((1 - x)^3 * (1 - x^3) * (1 - x^4)^2). - Michael Somos, May 04 2015
G.f.: f(x) - f(-x^2) where f(x) := x/(1-x)^2. - Michael Somos, May 04 2015
a(n) = -a(-n) for all n in Z. a(n) = n/2 * A068073(n).
a(n) = n*(4-i^n-(-i)^n)/4 with i=sqrt(-1). - Bruno Berselli, Mar 10 2011
a(n) = -(-1)^n * A186111(n). - Michael Somos, May 07 2015
a(n) = n - n*cos(n*Pi/2)/2. - Wesley Ivan Hurt, May 05 2021
Dirichlet g.f.: zeta(s-1) * (1 + 1/2^s - 1/4^(s-1)). - Amiram Eldar, Oct 26 2023
EXAMPLE
G.f. = x + 3*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 4*x^8 + 9*x^9 + ...
MATHEMATICA
CoefficientList[Series[x(1+x)(1+x^3)/((1-x)(1+x^2))^2, {x, 0, 80}], x] (* Harvey P. Dale, Mar 06 2011 *)
a[ n_] := n/2 {2, 3, 2, 1}[[ Mod[ n, 4, 1]]]; (* Michael Somos, May 04 2015 *)
PROG
(PARI) {a(n) = n/2 * [1, 2, 3, 2][n%4 + 1]};
(PARI) {a(n) = sign(n) * polcoeff( x * (1 + x) * (1 + x^3) / ((1 - x) * (1 + x^2))^2 + x * O(x^abs(n)), abs(n))};
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x)*(1+x^3)/((1-x)*(1+x^2))^2)); // G. C. Greubel, Aug 14 2018
CROSSREFS
Cf. A187601. - Bruno Berselli, Mar 12 2011
KEYWORD
nonn,easy,mult
AUTHOR
Michael Somos, Feb 27 2011
STATUS
approved