A186813 a(n) = n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.
0, 1, 3, 3, 2, 5, 9, 7, 4, 9, 15, 11, 6, 13, 21, 15, 8, 17, 27, 19, 10, 21, 33, 23, 12, 25, 39, 27, 14, 29, 45, 31, 16, 33, 51, 35, 18, 37, 57, 39, 20, 41, 63, 43, 22, 45, 69, 47, 24, 49, 75, 51, 26, 53, 81, 55, 28, 57, 87, 59, 30, 61, 93, 63, 32, 65, 99, 67, 34, 69, 105, 71, 36
Offset: 0
Examples
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 + ...
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).
Crossrefs
Cf. A187601. - Bruno Berselli, Mar 12 2011
Programs
-
Magma
m:=25; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x)*(1+x^3)/((1-x)*(1+x^2))^2)); // G. C. Greubel, Aug 14 2018 -
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 *) -
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))};
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