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 A014437 Odd Fibonacci numbers. 11
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Starting with offset 1 = row sums of triangle A177995. - Gary W. Adamson, May 16 2010 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 a={}; Do[f=Fibonacci[n]; If[OddQ[f], AppendTo[a, f]], {n, 1, 30, 1}]; a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *) 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 *) 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 CROSSREFS Cf. A001651, A059878, A000045. Cf. A177995. - Gary W. Adamson, May 16 2010 Sequence in context: A283752 A283602 A283817 * A153866 A104222 A240070 Adjacent sequences:  A014434 A014435 A014436 * A014438 A014439 A014440 KEYWORD nonn,easy AUTHOR EXTENSIONS a(30)-a(31) from Vincenzo Librandi, Apr 18 2011 STATUS approved

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