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A084179 Expansion of the g.f. x/((1+2x)(1-x-x^2)). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Sums of consecutive pairs yield A083178.

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(4Pi*k/m)Cos(2Pi*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

Table of n, a(n) for n=0..33.

Index to sequences with linear recurrences with constant coefficients, signature (-1,3,2).

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(4Pi*k/5)Cos(2Pi*k/5)^n). - Herbert Kociemba, May 31 2004

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Sequence in context: A037955 A225121 A225536 * A026634 A026656 A184244

Adjacent sequences:  A084176 A084177 A084178 * A084180 A084181 A084182

KEYWORD

easy,sign

AUTHOR

Paul Barry, May 18 2003

STATUS

approved

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Last modified December 22 17:45 EST 2014. Contains 252365 sequences.