|
|
A084179
|
|
Expansion of the g.f. x/((1+2x)(1-x-x^2)).
|
|
5
|
|
|
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
|
|
|
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):
|
|
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
|
|
|
KEYWORD
|
easy,sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|