A227200 a(n) = a(n-1) + a(n-2) - 2^(n-1) with a(0)=a(2)=0, a(1)=-a(3)=1, a(4)=-5.

0, 1, 0, -1, -5, -14, -35, -81, -180, -389, -825, -1726, -3575, -7349, -15020, -30561, -61965, -125294, -252795, -509161, -1024100, -2057549, -4130225, -8284926, -16609455, -33282989, -66669660, -133507081, -267285605, -535010414, -1070731475

Offset: 0

Author

Chandrakant N Phadte, Sep 18 2013

Keywords

• sign
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• C. N. Phadte and S. P. Pethe, On Second Order Non-homogeneous recurrence relation, Annales Mathematicae et informaticae, 41 (2013), pp. 205-210.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Crossrefs

Cf. versions with different signs: A027974, A142585.

Cf. A000032, A000045.

Programs

BASIC

LET N=0
LET L=0
LET M=1
PRINT L
PRINT M
FOR I=1 TO 30
LET N=M+L-(2)^(I-1)
PRINT N
LET L=M
LET M=N
NEXT I
END

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1-3*x)/((1-2*x)*(1-x-x^2)))); // Bruno Berselli, Oct 03 2013

Magma

I:=[0,1,0,-1,-5]; [n le 5 select I[n] else Self(n-1)+Self(n-2)-2^(n-3): n in [1..35]]; // Vincenzo Librandi, Oct 05 2013

Mathematica

Table[LucasL[n + 1] - 2^n, {n, 0, 30}] (* Bruno Berselli, Oct 03 2013 *)
CoefficientList[Series[x (1 - 3 x)/((1 - 2 x) (1 - x - x^2)), {x, 0, 40}], x](* Vincenzo Librandi, Oct 05 2013 *)

PARI

a(n)=fibonacci(n)+fibonacci(n+2)-2^n \\ Charles R Greathouse IV, Oct 03 2013

Formula

G.f.: x*(1-3*x)/((1-2*x)*(1-x-x^2)).

a(n) = -(-1)^n*A142585(n+1) = A000032(n+1) - 2^n. [Bruno Berselli, Oct 03 2013]

a(n) = 3*a(n-1) -a(n-2) -2*a(n-3). [Bruno Berselli, Oct 03 2013]

Extensions

More terms from Bruno Berselli, Oct 03 2013