A140359 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).

1, 1, 6, 11, 26, 51, 106, 211, 426, 851, 1706, 3411, 6826, 13651, 27306, 54611, 109226, 218451, 436906, 873811, 1747626, 3495251, 6990506, 13981011, 27962026, 55924051, 111848106, 223696211, 447392426, 894784851, 1789569706, 3579139411

Offset: 0

Author

Paul Curtz, Jun 24 2008

Keywords

• nonn

Comments

This is the sequence A(1,1;1,2;3) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

Cf. A000975, A001045, A087288, A137241, A140360.

Programs

Magma

[(5*2^(n+1) -9 + 5*(-1)^n)/6: n in [0..50]]; // G. C. Greubel, Oct 10 2017

Mathematica

Table[(5*2^(n+1) -9 + 5*(-1)^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 10 2017 *)
LinearRecurrence[{2,1,-2},{1,1,6},40] (* Harvey P. Dale, Mar 24 2021 *)

PARI

for(n=0,50, print1((5*2^(n+1) -9 + 5*(-1)^n)/6, ", ")) \\ G. C. Greubel, Oct 10 2017

Formula

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

a(n+1) - a(n) = 5*A001045(n), Jacobsthal numbers.

a(n+1) - 2*a(n) = (-1)^(n+1)* A010685(n).

From R. J. Mathar, Jul 10 2008: (Start)

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

a(n) = (5*2^(n+1) - 9 + 5*(-1)^n)/6. (End)

a(n) = a(n-1) + 2*a(n-2) +3, n>1 - Gary Detlefs, Jun 20 2010

Extensions

Extended by R. J. Mathar, Jul 10 2008