A014437 Odd Fibonacci numbers.
1, 1, 3, 5, 13, 21, 55, 89, 233, 377, 987, 1597, 4181, 6765, 17711, 28657, 75025, 121393, 317811, 514229, 1346269, 2178309, 5702887, 9227465, 24157817, 39088169, 102334155, 165580141, 433494437, 701408733, 1836311903, 2971215073, 7778742049, 12586269025
Offset: 0
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).
Programs
-
Magma
[Fibonacci((3*Floor((n+1)/2)) + (-1)^n): n in [0..50]]; // Vincenzo Librandi, Apr 18 2011
-
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 # second Maple program: a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|0|4|0>>^n.<<1,1,3,5>>)[1,1]: seq(a(n), n=0..33); # Alois P. Heinz, May 22 2025
-
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 *) -
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
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)
Extensions
a(30)-a(31) from Vincenzo Librandi, Apr 18 2011
Comments