OFFSET
0,3
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (0,4,0,1).
FORMULA
Fibonacci(3n+1) union Fibonacci(3n+2).
a(n) = Fibonacci(3*floor((n+1)/2) + (-1)^n). - Antti Karttunen, Feb 05 2001
G.f.: ( -1-x+x^2-x^3 ) / ( -1+4*x^2+x^4 ). - R. J. Mathar, Feb 16 2011
a(2n) = v-w, a(2n+1) = v+w, with v=A001076(n+1), w=A001076(n). Therefore, a(2n)+a(2n+1) = 2*A001076(n+1). - Ralf Stephan, Aug 31 2013
From Vladimir Reshetnikov, Oct 30 2015: (Start)
a(n) = ((cos(Pi*n/2)-sqrt(phi)*sin(Pi*n/2))/phi^((3*n+2)/2) + (sqrt(phi)*cos(Pi*n/2)^2+sin(Pi*n/2)^2)*phi^((3*n+1)/2))/sqrt(5), where phi=(1+sqrt(5))/2.
E.g.f.: (cos(x/phi^(3/2))/phi - sin(x/phi^(3/2))/sqrt(phi) + cosh(x*phi^(3/2))*phi + sinh(x*phi^(3/2))*sqrt(phi))/sqrt(5).
(End)
MAPLE
with(combinat):A014437:=proc(n)return fibonacci((3*floor((n+1)/2)) + (-1)^n):end:
seq(A014437(n), n=0..31); # Nathaniel Johnston, Apr 18 2011
MATHEMATICA
RecurrenceTable[{a[n] == 4*a[n-2] + a[n-4], a[0]==1, a[1]==1, a[2]==3, a[3]==5}, a, {n, 0, 500}] (* G. C. Greubel, Oct 30 2015 *)
Table[ SeriesCoefficient[(-1 - x + x^2 - x^3)/(-1 + 4*x^2 + x^4), {x, 0, n}], {n, 0, 20}] (* Nikolaos Pantelidis, Feb 01 2023 *)
Select[Fibonacci[Range[50]], OddQ] (* Harvey P. Dale, Sep 01 2023 *)
PROG
(Magma) [Fibonacci((3*Floor((n+1)/2)) + (-1)^n): n in [0..50]]; // Vincenzo Librandi, Apr 18 2011
(PARI) Vec((-1-x+x^2-x^3)/(-1+4*x^2+x^4) + O(x^200)) \\ Altug Alkan, Oct 31 2015
(PARI) apply( A014437(n)=fibonacci(n\/2*3+(-1)^n), [0..30]) \\ M. F. Hasler, Nov 18 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
a(30)-a(31) from Vincenzo Librandi, Apr 18 2011
STATUS
approved