OFFSET
0,3
COMMENTS
Fibonacci(2*n+1) repeated. a(n) is the right neighbor of Fibonacci(n+2) in A049456 and A002487 (starts 2,2,5,...). A000045(n+2) = A094966(n) + a(n).
Diagonal sums of A109223. - Paul Barry, Jun 22 2005
The Fi2 sums, see A180662, of triangle A065941 equal the terms of this sequence. - Johannes W. Meijer, Aug 11 2011
a(n) is the last term of (n+1)-th row in Wythoff array A003603. -Reinhard Zumkeller, Jan 26 2012
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).
FORMULA
G.f.: (1+x-x^2-x^3)/(1-3*x^2+x^4).
a(n) = Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2)+k, 2*k). - Paul Barry, Jun 22 2005
Starting (1, 2, 2, 5, 5, 13, 13, ...) = A133080 * A000045, where A000045 starts with "1". - Gary W. Adamson, Sep 08 2007
a(n) = Fibonacci(n+1)^(4*k+3) mod Fibonacci(n+2), for any k>-1, n>0. - Gary Detlefs, Nov 29 2010
MATHEMATICA
LinearRecurrence[{0, 3, 0, -1}, {1, 1, 2, 2}, 40] (* Harvey P. Dale, Apr 05 2015 *)
f[n_]:=If[OddQ@n, (Fibonacci[n]), Fibonacci[n+1]]; Array[f, 100, 0] (* Vincenzo Librandi, Nov 18 2018 *)
Table[Fibonacci[n, 0]*Fibonacci[n] + LucasL[n, 0]*Fibonacci[n + 1]/2, {n, 0, 50}] (* G. C. Greubel, Nov 18 2018 *)
PROG
(Magma) [IsEven(n) select Fibonacci(n+1) else Fibonacci(n): n in [0..70]]; // Vincenzo Librandi, Nov 18 2018
(PARI) vector(50, n, n--; fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2) \\ G. C. Greubel, Nov 18 2018
(Sage) [fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2 for n in range(50)] # G. C. Greubel, Nov 18 2018
(GAP) List([0..50], n -> Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2); # G. C. Greubel, Nov 18 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 26 2004
STATUS
approved