login
A014437
Odd Fibonacci numbers.
16
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
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. A360957 (sum of reciprocals).
Sequence in context: A283602 A283817 A340400 * A153866 A104222 A309703
KEYWORD
nonn,easy
EXTENSIONS
a(30)-a(31) from Vincenzo Librandi, Apr 18 2011
STATUS
approved