OFFSET
0,2
COMMENTS
Let M = an infinite triangle with (1,2,2,3,3,4,4,...) as the left border and all other columns = (0,1,2,3,4,5,...). Then lim_{n->infinity} M^n = A099016, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 26 2010
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
FORMULA
G.f.: (1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2)).
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = 2*F(n)^2 + F(n)*F(n-1) + F(n-1)^2, where F = A000045.
a(n) = 3*((3/2 - sqrt(5)/2)^n + (3/2 + sqrt(5)/2)^n)/5 - (-1)^n/5.
a(n) = A099015(n)/F(n+1).
a(n) = 3*A005248(n)/5 - (-1)^n/5.
a(n) = 3*A000032(2*n)/5 - (-1)^n/5.
a(n) = A061646(n) + F(n)^2.
a(n) = 3*F(n)^2 + (-1)^n.
a(n) = F(n+1)^2 + F(n)*F(n-2). See also A192914, fourth formula. - Bruno Berselli, Feb 15 2017
MAPLE
with(combinat):seq(3*fibonacci(n)^2+(-1)^n, n= 0..27)
MATHEMATICA
CoefficientList[Series[(1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 31 2017 *)
PROG
(Magma) [3*Lucas(2*n)/5-(-1)^n/5: n in [0..35]]; // Vincenzo Librandi, Jun 09 2011
(Magma) F:=Fibonacci; [F(n+1)^2+F(n)*F(n-2): n in [0..30]]; // Bruno Berselli, Feb 15 2017
(PARI) x='x+O('x^30); Vec((1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2))) \\ G. C. Greubel, Dec 31 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Sep 22 2004
STATUS
approved