A132353 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), starting with 1, 2, 6, 20.

1, 2, 6, 20, 61, 183, 547, 1640, 4920, 14762, 44287, 132861, 398581, 1195742, 3587226, 10761680, 32285041, 96855123, 290565367, 871696100, 2615088300, 7845264902, 23535794707, 70607384121, 211822152361, 635466457082

Offset: 0

Author

Paul Curtz, Nov 24 2007

Keywords

• nonn

Comments

A132868(n) - a(n) = A128834(n) (discovered in 1995).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,0,-1,3).

Crossrefs

Cf. A129339.

Programs

Magma

I:=[1,2,6,20]; [n le 4 select I[n] else 3*Self(n-1) - Self(n-3) + 3*Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018

Mathematica

LinearRecurrence[{3, 0, -1, 3}, {1, 2, 6, 20}, 50] (* G. C. Greubel, Jan 15 2018 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,3d-b+3a}; NestList[nxt,{1,2,6,20},50][[;;,1]] (* Harvey P. Dale, Feb 17 2025 *)

PARI

x='x+O('x^30); Vec((1-x+3*x^3)/((1-3*x)*(1+x)*(x^2-x+1))) \\ G. C. Greubel, Jan 15 2018

Formula

Also a(n) - 3^(n+1) = hexaperiodic 1, -1, -3, -1, 1, 3; cf. A132951.

From R. J. Mathar, Apr 04 2008: (Start)

O.g.f.: (1-x+3*x^3)/((1-3*x)*(1+x)*(x^2-x+1)).

a(n) = -(-1)^n/12 + 3^(n+1)/4 + A057079(n+2)/3. (End)

Extensions

More terms from R. J. Mathar, Apr 04 2008