OFFSET
0,3
COMMENTS
The g.f. is obtained from that of A003462 through the mapping g(x)->g(x(1-x)). A003462 may be retrieved through the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. Binomial transform of x(1+x)/(1+x^2+x^4).
The sequence is identical to its sixth differences. See A140344. - Paul Curtz, Nov 09 2012
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).
FORMULA
G.f.: x(1-x)/((1-x+x^2)*(1-3*x+3*x^2));
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(3^(n-k)-1)/2.
a(n) = (1/2)*[A057083(n) - [1,1,0,0,-1,-1]_6 ]. - Ralf Stephan, Nov 15 2010
a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 3*a(n-4) = A140343(n+2) - A140343(n+1). - Paul Curtz, Nov 09 2012
a(n) is the binomial transform of the sequence 0, 1, 1, -1, -1, 0, ... = A103368(n+5). - Paul Curtz, Nov 09 2012
EXAMPLE
From Paul Curtz, Nov 09 2012: (Start)
The sequence and its higher-order differences (periodic after 6 rows):
0, 1, 3, 5, 5, 0, -14, ...
1, 2, 2, 0, -5, -14, -27, ...
1, 0, -2, -5, -9, -13, -13, ...
-1, -2, -3, -4, -4, 0, 13, ... = -A134581(n+1)
-1, -1, -1, 0, 4, 13, 27, ...
0, 0, 1, 4, 9, 14, 14, ... = A140343(n+2)
0, 1, 3, 5, 5, 0, -14, ...
(End)
MATHEMATICA
LinearRecurrence[{4, -7, 6, -3}, {0, 1, 3, 5}, 35] (* Vincenzo Librandi, Dec 24 2018 *)
PROG
(Magma) I:=[0, 1, 3, 5]; [n le 4 select I[n] else 4*Self(n-1)-7*Self(n-2)+ 6*Self(n-3)-3*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 24 2018
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Apr 26 2005
STATUS
approved