OFFSET
0,2
COMMENTS
If a(0) is put to 2 instead of 1 this becomes a(n) = (-1)^n*A005248(n), n >= 0. These are then the alternating row sums of triangle A127677.
Also abs(a(n)) is the number of rounded area of pentagon or pentagram in series arrangement. - Kival Ngaokrajang, Mar 27 2013
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 8.
Kival Ngaokrajang, Pentagram for n = 1..6
Eric Weisstein's World of Mathematics, Pentagram
Index entries for linear recurrences with constant coefficients, signature (-3,-1).
FORMULA
a(0) = 1 and a(n) = (-1)^n*(F(2*(n+1)) - F(2*(n-1))) = (-1)^n*L(2*n), n>=1, with F=A000045 (Fibonacci) and L=A000032 (Lucas).
O.g.f.: (1-x^2)/(1+3*x+x^2).
G.f.: (W(0) -6)/(5*x) -1 , where W(k) = 5*x*k + x + 6 - 6*x*(5*k-9)/W(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
From Colin Barker, Oct 14 2015: (Start)
a(n) = -3*a(n-1) - a(n-2) for n>2.
a(n) = (1/2*(-3-sqrt(5)))^n+(1/2*(-3+sqrt(5)))^n for n>0.
(End)
E.g.f.: 2*exp(-3*x/2)*cosh(sqrt(5)*x/2) - 1. - Stefano Spezia, Dec 26 2021
PROG
(PARI) Vec((1-x^2)/(1+3*x+x^2) + O(x^40)) \\ Colin Barker, Oct 14 2015
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Nov 16 2012
STATUS
approved