A152733 a(n) + a(n+1) + a(n+2) = 3^n.
0, 0, 3, 6, 18, 57, 168, 504, 1515, 4542, 13626, 40881, 122640, 367920, 1103763, 3311286, 9933858, 29801577, 89404728, 268214184, 804642555, 2413927662, 7241782986, 21725348961, 65176046880, 195528140640, 586584421923, 1759753265766, 5279259797298
Offset: 1
Keywords
Examples
0 + 0 + 3 = 3^1; 0 + 3 + 6 = 3^2; 3 + 6 + 18 = 3^3; ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (2, 2, 3).
Programs
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Magma
[n le 2 select 0 else 3^(n-2) -Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 31 2014
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Mathematica
k0=k1=0;lst={k0,k1};Do[kt=k1;k1=3^n-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst Rest[CoefficientList[Series[3x^3/((1-3x)(1+x+x^2)),{x,0,30}],x]] (* Harvey P. Dale, Aug 31 2014 *) -
PARI
x='x+O('x^30); concat([0,0], Vec(3*x^3/((1-3*x)*(1+x+x^2)))) \\ G. C. Greubel, Sep 01 2018
Formula
From R. J. Mathar, Dec 12 2008: (Start)
a(n) = 3*A077834(n-3).
G.f.: 3*x^3/((1-3*x)*(1+x+x^2)). (End)
a(n) = (1/13)*(3^n + 12*cos((2*n*Pi)/3) + 2*sqrt(3)*sin((2*n*Pi)/3)), n=1,2,... - Zak Seidov, Dec 12 2008