OFFSET
0,2
COMMENTS
In general, the sequence with g.f. 1/(1-2r*x+(r^2+1)*x^2) = 1/((1-r*x)^2+x^2) has a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*(r^2-1)^k*(2r)^(n-2k); a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+1,2k+1)*(-1)^k*r^(n-2k).
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-17).
FORMULA
G.f.: 1/((1-4*x)^2+x^2).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-17)^k*8^(n-2k).
a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+1, 2k+1)*(-1)^k*4^(n-2k).
a(n) = 8*a(n-1) - 17*a(n-2), n >= 2. - Vincenzo Librandi, Mar 18 2011
a(n) = (1/2-2*i)*(4+i)^n + (1/2+2*i)*(4-i)^n, where i is the imaginary unit. - Gerry Martens, Mar 19 2024
E.g.f.: exp(4*x) * (cos(x) + 4*sin(x)). - Amiram Eldar, Feb 10 2026
MATHEMATICA
Join[{a=1, b=8}, Table[c=8*b-17*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 20 2011 *)
LinearRecurrence[{8, -17}, {1, 8}, 30] (* Harvey P. Dale, Mar 24 2025 *)
PROG
(SageMath) [lucas_number1(n, 8, 17) for n in range(1, 26)] # Zerinvary Lajos, Apr 23 2009
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, May 01 2005
STATUS
approved