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A014437 Odd Fibonacci numbers.

%I A014437 #72 May 22 2025 18:06:52
%S A014437 1,1,3,5,13,21,55,89,233,377,987,1597,4181,6765,17711,28657,75025,
%T A014437 121393,317811,514229,1346269,2178309,5702887,9227465,24157817,
%U A014437 39088169,102334155,165580141,433494437,701408733,1836311903,2971215073,7778742049,12586269025
%N A014437 Odd Fibonacci numbers.
%H A014437 Nathaniel Johnston, <a href="/A014437/b014437.txt">Table of n, a(n) for n = 0..500</a>
%H A014437 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,1).
%F A014437 Fibonacci(3n+1) union Fibonacci(3n+2).
%F A014437 a(n) = Fibonacci(3*floor((n+1)/2) + (-1)^n). - _Antti Karttunen_, Feb 05 2001
%F A014437 G.f.: ( -1-x+x^2-x^3 ) / ( -1+4*x^2+x^4 ). - _R. J. Mathar_, Feb 16 2011
%F A014437 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
%F A014437 From _Vladimir Reshetnikov_, Oct 30 2015: (Start)
%F A014437 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.
%F A014437 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).
%F A014437 (End)
%p A014437 with(combinat):A014437:=proc(n)return fibonacci((3*floor((n+1)/2)) + (-1)^n):end:
%p A014437 seq(A014437(n),n=0..31); # _Nathaniel Johnston_, Apr 18 2011
%p A014437 # second Maple program:
%p A014437 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]:
%p A014437 seq(a(n), n=0..33);  # _Alois P. Heinz_, May 22 2025
%t A014437 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 *)
%t A014437 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 *)
%t A014437 Select[Fibonacci[Range[50]],OddQ] (* _Harvey P. Dale_, Sep 01 2023 *)
%o A014437 (Magma) [Fibonacci((3*Floor((n+1)/2)) + (-1)^n): n in [0..50]]; // _Vincenzo Librandi_, Apr 18 2011
%o A014437 (PARI) Vec((-1-x+x^2-x^3)/(-1+4*x^2+x^4) + O(x^200)) \\ _Altug Alkan_, Oct 31 2015
%o A014437 (PARI) apply( A014437(n)=fibonacci(n\/2*3+(-1)^n), [0..30]) \\ _M. F. Hasler_, Nov 18 2018
%Y A014437 Cf. A001651, A059878, A000045.
%Y A014437 Cf. A360957 (sum of reciprocals).
%K A014437 nonn,easy
%O A014437 0,3
%A A014437 _Mohammad K. Azarian_
%E A014437 a(30)-a(31) from _Vincenzo Librandi_, Apr 18 2011