A084179 Expansion of the g.f. x/((1+2x)(1-x-x^2)).

0, 1, -1, 4, -5, 15, -22, 57, -93, 220, -385, 859, -1574, 3381, -6385, 13380, -25773, 53143, -103702, 211585, -416405, 843756, -1669801, 3368259, -6690150, 13455325, -26789257, 53774932, -107232053, 214978335, -429124630, 859595529, -1717012749, 3437550076

Offset: 0

Author

Paul Barry, May 18 2003

Keywords

• easy
• sign

Comments

Sums of consecutive pairs yield A084178.

Number of walks of length n+1 between two vertices at distance 2 in the cycle graph C_5. In general a(n,m) = 2^n/m*Sum_{k=0..m-1} cos(4*Pi*k/m)*cos(2*Pi*k/m)^n is the number of walks of length n between two vertices at distance 2 in the cycle graph C_m. - Herbert Kociemba, May 31 2004

Links

• Robert Israel, Table of n, a(n) for n = 0..3260
• Index entries for linear recurrences with constant coefficients, signature (-1,3,2).

Crossrefs

Cf. A000032, A007598, A084178.

Programs

Magma

I:=[0,1,-1]; [n le 3 select I[n] else -Self(n-1)+3*Self(n-2)+2*Self(n-3): n in [1..45]]; // Vincenzo Librandi, Nov 10 2014

Maple

f:= gfun:-rectoproc({a(n) = -a(n-1)+3*a(n-2)+2*a(n-3),
   a(0)=0,a(1)=1,a(2)=-1},a(n),remember):
seq(f(n), n=0..100); # Robert Israel, Dec 11 2015

Mathematica

CoefficientList[Series[x / ((1 + 2 x) (1 - x - x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)

PARI

concat(0, Vec(x/((1+2*x)*(1-x-x^2)) + O(x^100))) \\ Altug Alkan, Dec 11 2015

Formula

Binomial transform is A007598. The unsigned sequence has G.f. x/((1-2x)(1+x-x^2)) with a(n) = 2*2^n/5-(-1)^n*A000032(n)/5. - Paul Barry, Apr 17 2004

a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n, k)*Fib(k)^2; a(n) = ((1/2-sqrt(5)/2)^n+(1/2+sqrt(5)/2)^n-2(-2)^n)/5; a(n) = A000032(n)/5-2(-2)^n/5. - Paul Barry, Apr 17 2004

a(n) = 2^n/5*Sum_{k=0..4} cos(4*Pi*k/5)*cos(2*Pi*k/5)^n. - Herbert Kociemba, May 31 2004

a(n) = -a(n-1) + 3*a(n-2) + 2*a(n-3) for n>2. - Paul Curtz, Mar 09 2008