A167193 a(n) = (1/3)*(1 - (-2)^n + 3*(-1)^n ) = (-1)^(n+1)*A167030(n).

1, 0, 0, 2, -4, 10, -20, 42, -84, 170, -340, 682, -1364, 2730, -5460, 10922, -21844, 43690, -87380, 174762, -349524, 699050, -1398100, 2796202, -5592404, 11184810, -22369620, 44739242, -89478484, 178956970, -357913940, 715827882, -1431655764, 2863311530, -5726623060, 11453246122, -22906492244, 45812984490

Offset: 0

Author

Paul Curtz, Oct 30 2009

Keywords

• easy
• sign

Comments

This is the inverse binomial transform of 1, 1, 1, 3, 5, 11,.. (continued as in A001045 and conjectured to be equal to A152046).

Any sequence (like this one) which obeys a(n)= -2a(n-1)+a(n-2)+2a(n-3) also obeys a(n)=5a(n-2)-4a(n-4), proved by telescoping; see A101622.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,1,2).

Programs

Magma

[( 1-(-1)^n*2^n)/3+(-1)^n: n in [0..40] ]; // Vincenzo Librandi, Aug 06 2011

Mathematica

LinearRecurrence[{-2,1,2},{1,0,0}, 25] (* or *) Table[(1/3)*(1 + 3*(-1)^n - (-2)^n), {n,0,25}] (* G. C. Greubel, Jun 04 2016 *)

Formula

a(2n) = (-1)^n* A084240(n). a(2n+1) = A020988(n).

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

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

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

E.g.f.: (1/3)*(exp(x) + 3*exp(-x) - exp(-2*x)). - G. C. Greubel, Jun 04 2016