A115335 a(0) = 3, a(1) = 5, a(2) = 1, and a(n) = (2^(1 + n) - 11*(-1)^n)/3 for n > 2.

3, 5, 1, 9, 7, 25, 39, 89, 167, 345, 679, 1369, 2727, 5465, 10919, 21849, 43687, 87385, 174759, 349529, 699047, 1398105, 2796199, 5592409, 11184807, 22369625, 44739239, 89478489, 178956967, 357913945, 715827879, 1431655769, 2863311527, 5726623065, 11453246119

Offset: 0

Author

Roger L. Bagula, Mar 06 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2).

Crossrefs

Cf. A115113, A115164, A163868 (bisection).

Programs

GAP

a:=[3,5,1];;  for n in [4..35] do a[n]:=(2^n-11*(-1)^(n-1))/3; od; a; # Muniru A Asiru, Nov 23 2018

Magma

I:=[9,7]; [3,5,1] cat [n le 2 select I[n] else Self(n-1) + 2*Self(n-2): n in [1..45]]; // G. C. Greubel, Nov 23 2018

Maple

seq(coeff(series((4*x^4+2*x^3+10*x^2-2*x-3)/((x+1)*(2*x-1)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Nov 23 2018

Mathematica

Join[{3,5,1},LinearRecurrence[{1,2},{9,7},40]] (* Harvey P. Dale, Jul 17 2014 *)

Maxima

append([3, 5, 1], makelist((2^(1 + n) - 11*(-1)^n)/3, n, 3, 40)); /* Franck Maminirina Ramaharo, Nov 23 2018 */

PARI

my(x='x+O('x^50)); Vec((4*x^4+2*x^3+10*x^2-2*x-3)/((x+1)*(2*x-1))) \\ G. C. Greubel, Nov 23 2018

Sage

s=((4*x^4+2*x^3+10*x^2-2*x-3)/((x+1)*(2*x-1))).series(x,50); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 23 2018

Formula

a(n) = 2*abs((1/2)*(-1 + (-2)^n) - (2/3)*(2 + (-2)^n)*A057427(n)).

From Colin Barker, Jan 04 2013: (Start)

a(n) = a(n-1) + 2*a(n-2) for n > 4.

G.f.: (4*x^4 + 2*x^3 + 10*x^2 - 2*x - 3) / ((x + 1)*(2*x - 1)). (End)

E.g.f.: (18 + 3*x^2 - 11*exp(-x) + 2*exp(2*x))/3. - Franck Maminirina Ramaharo, Nov 23 2018

Extensions

a(24) corrected, new name, and editing by Colin Barker and Joerg Arndt, Jan 04 2013