A180964 a(0)=1; for n>0, a(n) = 1 + 3*A117571(n-1).

1, 4, 4, 10, 13, 13, 19, 22, 22, 28, 31, 31, 37, 40, 40, 46, 49, 49, 55, 58, 58, 64, 67, 67, 73, 76, 76, 82, 85, 85, 91, 94, 94, 100, 103, 103, 109, 112, 112, 118, 121, 121, 127, 130, 130, 136, 139, 139, 145, 148, 148, 154, 157, 157, 163, 166, 166, 172

Offset: 0

Author

Bruno Berselli, Sep 28 2010 - Oct 01 2010

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A061347.

Programs

Magma

I:=[1, 4, 4, 10]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Mar 26 2013

Maple

m:=60; S:=series( (1+3*x+5*x^3)/((1-x)^2*(1+x+x^2)), x, m+1):
seq(coeff(S, x, j), j=1..m); # G. C. Greubel, Apr 06 2021

Mathematica

CoefficientList[Series[(1 +3x +5x^3)/((1-x)^2(1+x+x^2)), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{1,0,1,-1},{1,4,4,10},60] (* Harvey P. Dale, Aug 05 2020 *)

Sage

[3*n +chebyshev_U(n, -1/2) +2*chebyshev_U(n-1, -1/2) for n in (0..60)] # G. C. Greubel, Apr 06 2021

Formula

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

a(n) = a(n-1) +a(n-3) -a(n-4) for n>3.

a(n) = (n-1)*(n mod 3) +(n+1)*(n+1 mod 3) +n*(n+2 mod 3).

a(n) = 3*n +sqrt(3)*cos((4*n-3)*Pi/6) -sin((4*n-3)*Pi/6).

a(n) - a(n-1) = 3*A080425(n+1) for n>0.

From G. C. Greubel, Apr 06 2021: (Start)

a(n) = 3*n - 2*cos(2*Pi*(n+1)/3) = 3*n + A061347(n+1).

a(n) = 3*n + ChebyshevU(n, -1/2) + 2*ChebyshevU(n-1, -1/2). (End)