OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..2500
E. S. Egge and T. Mansour, Restricted permutations, Fibonacci numbers and k-generalized Fibonacci numbers, arXiv:math/0203226 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
FORMULA
a(0) = 0, a(1) = 1, a(n) = a(n-1) + a(n-2) + 3.
G.f.: x*(1+2*x)/((1-x)*(1-x-x^2)). Differences of A023537. - Ralf Stephan, Feb 07 2004
a(n) = A101220(3, 1, n). - Ross La Haye, Jan 28 2005
a(n) = F(n) + F(n+2) - 3, n >= 2, where F(n) is the n-th Fibonacci number. - Zerinvary Lajos, Jan 31 2008
a(n) = Sum_{k=1..n} ((-1/phi)^k + (phi)^k) where phi = 1/2+1/2*sqrt(5). - Dimitri Papadopoulos, Jan 07 2016
a(n) = 2*a(n-1)-a(n-3) for n>3. - Wesley Ivan Hurt, Jan 07 2016
MAPLE
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2]+3 od: seq(a[n], n=1..40); # Miklos Kristof, Mar 09 2005
with(combinat): seq(fibonacci(n)+fibonacci(n+2)-3, n=2..40); # Zerinvary Lajos, Jan 31 2008
g:=(1+z^2)/(1-z-z^2): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-3, n=3..40); # Zerinvary Lajos, Jan 09 2009
MATHEMATICA
LucasL[Range[3, 40]] - 3 (* Alonso del Arte, Sep 26 2013 *)
PROG
(Magma) [Lucas(n+2)-3: n in [1..40]]; // Vincenzo Librandi, Apr 16 2011
(PARI) vector(40, n, fibonacci(n+3) +fibonacci(n+1) -3) \\ G. C. Greubel, Dec 18 2017
(PARI) first(n) = Vec(x*(1+2*x)/((1-x)*(1-x-x^2)) + O(x^(n+1))) \\ Iain Fox, Dec 19 2017
(Sage) [lucas_number2(n+2, 1, -1) -3 for n in (1..40)] # G. C. Greubel, Jun 01 2019
(GAP) List([1..40], n-> Lucas(1, -1, n+2)[2] -3 ) # G. C. Greubel, Jun 01 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved