OFFSET
0,2
COMMENTS
The generic a(n) = 2*a(n-1)-a(n-3) for this family of recurrences (see the link to the OEIS index) leads directly to a common symmetry of the form a(n+1)-2a(n) = 12, 10, 0, -12, -34, -68, -124,... = 12, 10, -a(n).
LINKS
Todd Silvestri, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
FORMULA
G.f.: 2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)).
a(n) = 2*A022095(n+3) - 22. - R. J. Mathar, Jul 07 2011
a(n) = 23*F(n)+11*L(n)-22 = 23*A000045(n)+11*A000032(n)-22, where F(n) and L(n) are the n-th Fibonacci and Lucas numbers, respectively. - Todd Silvestri, Dec 16 2014
a(n) = (1/5)*(-110 + (55-23*sqrt(5))*((1-sqrt(5))/2)^n + ((1+sqrt(5))/2)^n*(55+23*sqrt(5))). - Colin Barker, Nov 13 2017
MATHEMATICA
a[n_Integer/; n>=0]:=23 Fibonacci[n]+11 LucasL[n]-22 (* Todd Silvestri, Dec 16 2014 *)
LinearRecurrence[{2, 0, -1}, {0, 12, 34}, 40] (* Harvey P. Dale, May 12 2015 *)
PROG
(PARI) concat(0, Vec(2*x*(6 + 5*x) / ((1 - x)*(1 - x - x^2)) + O(x^50))) \\ Colin Barker, Nov 13 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Sep 18 2008
STATUS
approved