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 A190949 Odd Fibonacci numbers with odd index. 2
 1, 5, 13, 89, 233, 1597, 4181, 28657, 75025, 514229, 1346269, 9227465, 24157817, 165580141, 433494437, 2971215073, 7778742049, 53316291173, 139583862445, 956722026041, 2504730781961, 17167680177565, 44945570212853, 308061521170129, 806515533049393 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS All prime Fibonacci numbers (A005478) except 2 and 3 are in this sequence. All terms equal 1 (mod 4). The indices of these Fibonacci numbers are 6k-1 or 6k+1. This sequence can be thought of as two interlocking sequences, each of the form a(n) = 18a(n - 1) - a(n - 2). Proof that all terms equal 1 (mod 4): From the Lucas 1876 identity Fib(2n+1) = Fib(n)^2 + Fib(n+1)^2 (see Weisstein, formula 60, or page 79 of Koshy), we see that odd-indexed Fibonacci numbers are the sum of two squares. Because a square is 0 or 1 (mod 4), the sum of two squares is 0, 1, or 2 (mod 4). All these terms are odd numbers. Hence, the only possibility is that they are 1 (mod 4). This can also be proved from the recursion formula. REFERENCES Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001. LINKS T. D. Noe, Table of n, a(n) for n = 1..200 Eric W. Weisstein, Fibonacci Number Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1). FORMULA a(n) = 18*a(n-2) - a(n-4), with a(1)=1, a(2)=5, a(3)=13, and a(4)=89. G.f.: x*(1-x)*(1+6*x+x^2)/((1+4*x-x^2)*(1-4*x-x^2)). - Colin Barker, Jun 19 2012 a(n) = (-(2-sqrt(5))^n + (-2-sqrt(5))^n*(-2+sqrt(5)) + 2*(-2+sqrt(5))^n + sqrt(5)*(-2+sqrt(5))^n + (2+sqrt(5))^n)/(2*sqrt(5)) for n>0. - Colin Barker, Mar 27 2016 MATHEMATICA LinearRecurrence[{0, 18, 0, -1}, {1, 5, 13, 89}, 50] PROG (PARI) a(n)=fibonacci(n\2*6+if(n%2, 1, -1)) \\ Charles R Greathouse IV, Jun 08 2011 (PARI) Vec(x*(1-x)*(1+6*x+x^2)/((1+4*x-x^2)*(1-4*x-x^2)) + O(x^30)) \\ Colin Barker, Mar 27 2016 CROSSREFS Cf. A000045 (Fibonacci numbers). Sequence in context: A359316 A165262 A092955 * A263468 A350467 A081560 Adjacent sequences: A190946 A190947 A190948 * A190950 A190951 A190952 KEYWORD nonn,easy AUTHOR T. D. Noe, May 24 2011 STATUS approved

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Last modified April 23 15:20 EDT 2024. Contains 371916 sequences. (Running on oeis4.)