OFFSET
0,2
COMMENTS
Row sums of triangle A135222. - Gary W. Adamson, Nov 23 2007
a(n) is the F(n+1)-th highest positive integer not equal to any a(k), 1 <= k <= n-1, where F(n) = Fibonacci numbers = A000045(n). - Jaroslav Krizek, Oct 28 2009
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).
FORMULA
a(n) = (sqrt(5)*(1+sqrt(5))^(n+1) - sqrt(5)*(1-sqrt(5))^(n+1))/(10*2^n) + n.
G.f.: (1-x-x^3)/((1-x-x^2)*(1-x)^2).
From Jaroslav Krizek, Oct 28 2009: (Start)
a(0) = 1, a(n) = a(n-1) + A000045(n-1) + 1 for n >= 1.
a(0) = 1, a(1) = 2, a(2) = 4, a(n) = a(n-1) + a(n-2) - (n-3) n >= 3. (End)
E.g.f.: (1/10)*exp(-2*x/(1+sqrt(5)))*(5 - sqrt(5) + (5 + sqrt(5))*exp(sqrt(5)*x) + 10*exp((1/2)*(1+sqrt(5))*x)*x). - Stefano Spezia, Nov 20 2019
MAPLE
with(combinat); seq(n + fibonacci(n+1), n=0..40); # G. C. Greubel, Nov 20 2019
MATHEMATICA
Table[ Fibonacci[n+1]+n, {n, 0, 38}] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011 *)
CoefficientList[Series[(x^3+x-1)/((x-1)^2 (x^2+x-1)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 10 2013 *)
LinearRecurrence[{3, -2, -1, 1}, {1, 2, 4, 6}, 40] (* Harvey P. Dale, Mar 02 2016 *)
PROG
(MuPAD) numlib::fibonacci(n)+n-1 $ n = 1..48; // Zerinvary Lajos, May 08 2008
(Magma) [n+Fibonacci(n+1): n in [0..40]]; // Vincenzo Librandi, Aug 10 2013
(PARI) a(n)=n+fibonacci(n) \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n+fibonacci(n+1) for n in range(40)] # G. C. Greubel, Feb 12 2019
(GAP) List([0..40], n-> n + Fibonacci(n+1) ); # G. C. Greubel, Nov 20 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Mar 26 2003
STATUS
approved