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 A084179 Expansion of the g.f. x/((1+2x)(1-x-x^2)). 4
 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 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). 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 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 *) 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 (PARI) concat(0, Vec(x/((1+2*x)*(1-x-x^2)) + O(x^100))) \\ Altug Alkan, Dec 11 2015 CROSSREFS Cf. A000032, A007598, A084178. Sequence in context: A225121 A267991 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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