A152731 -id:A152731 - cp's OEIS frontend

A152732 a(n) + a(n+1) + a(n+2) = 2^n.

0, 0, 2, 2, 4, 10, 18, 36, 74, 146, 292, 586, 1170, 2340, 4682, 9362, 18724, 37450, 74898, 149796, 299594, 599186, 1198372, 2396746, 4793490, 9586980, 19173962, 38347922, 76695844, 153391690, 306783378, 613566756, 1227133514, 2454267026, 4908534052

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

0 + 0 + 2 = 2^1; 0 + 2 + 2 = 2^2; 2 + 2 + 4 = 2^3; 2 + 4 + 10 = 2^4; ...

With a(0)=1, a(n) is the number of length n strings in the language over alphabet {0,1} generated by the regular expression: ((0+1)(0*(11)*)*10)*. - Geoffrey Critzer, Jan 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A152728, A152729, A152730, A152731, A152725, A152726, A000212.

I:=[0,0,2]; [n le 3 select I[n] else Self(n-1) +Self(n-2) +2*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 01 2018

k0=k1=0;lst={k0,k1};Do[kt=k1;k1=2^n-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
LinearRecurrence[{1, 1, 2}, {0, 0, 2}, 70] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)

concat([0,0],Vec(2/(1-2*x)/(1+x+x^2)+O(x^99))) \\ Charles R Greathouse IV, Feb 24 2012

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = 2*A077947(n-3).
G.f.: 2*x^3/((1-2*x)*(1+x+x^2)). (End)
a(n) = (1/21)*(3*2^n + 18*cos((2*n*Pi)/3) + 2*sqrt(3)*sin((2*n*Pi)/3)). - Zak Seidov, Dec 12 2008

A152733 a(n) + a(n+1) + a(n+2) = 3^n.

Original entry on oeis.org

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

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

nonn

			0 + 0 + 3 = 3^1; 0 + 3 + 6 = 3^2; 3 + 6 + 18 = 3^3; ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 2, 3).

Cf. A152728, A152729, A152730, A152731, A152732, A152725, A152726, A000212.

[n le 2 select 0 else 3^(n-2) -Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 31 2014

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 *)

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

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

cp's OEIS Frontend

A152732 a(n) + a(n+1) + a(n+2) = 2^n.

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