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

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

Cf. A001651, A059878, A000045.

Cf. A360957 (sum of reciprocals).

Sequence in context: A283602 A283817 A340400 * A153866 A104222 A309703

Adjacent sequences: A014434 A014435 A014436 * A014438 A014439 A014440

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

a(30)-a(31) from Vincenzo Librandi, Apr 18 2011

Status

approved