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A167816 Numerator of x(n) = x(n-1) + x(n-2), x(0)=0, x(1)=1/3; denominator=A167817.

%I A167816 #23 Sep 08 2022 08:45:48
%S A167816 0,1,1,2,1,5,8,13,7,34,55,89,48,233,377,610,329,1597,2584,4181,2255,
%T A167816 10946,17711,28657,15456,75025,121393,196418,105937,514229,832040,
%U A167816 1346269,726103,3524578,5702887,9227465,4976784,24157817,39088169,63245986,34111385
%N A167816 Numerator of x(n) = x(n-1) + x(n-2), x(0)=0, x(1)=1/3; denominator=A167817.
%H A167816 Vincenzo Librandi, <a href="/A167816/b167816.txt">Table of n, a(n) for n = 0..1000</a>
%H A167816 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%H A167816 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 7, 0, 0, 0, -1).
%F A167816 a(n) = (a(n-1)*A093148(n+2) + a(n-2)*A093148(n+1))/A093148(n-1) for n>1.
%F A167816 a(4*n) = A004187(n) = (a(4*n-1) + a(4*n-2))/3;
%F A167816 a(4*n+1) = A033889(n) = 3*a(4*n-1) + a(4*n-2);
%F A167816 a(4*n+2) = A033890(n) = a(4*n-1) + 3*a(4*n-2);
%F A167816 a(4*n+3) = A033891(n) = a(4*n-1) + a(4*n-2).
%F A167816 Numerator of Fibonacci(n) / Fibonacci(2n-4) for n>=3. - _Gary Detlefs_, Dec 20 2010
%t A167816 Numerator[LinearRecurrence[{1,1},{0,1/3},40]] (* _Harvey P. Dale_, Dec 07 2014 *)
%t A167816 LinearRecurrence[{0, 0, 0, 7, 0, 0, 0, -1},{0, 1, 1, 2, 1, 5, 8, 13},39] (* _Ray Chandler_, Aug 03 2015 *)
%o A167816 (Magma) [0,1,1] cat [Numerator(Fibonacci(n)/Fibonacci(2*n-4)): n in [3..40]]; // _Vincenzo Librandi_, Jun 28 2016
%Y A167816 Cf. A000045, A167808.
%K A167816 frac,nonn
%O A167816 0,4
%A A167816 _Reinhard Zumkeller_, Nov 13 2009
%E A167816 Definition corrected by _D. S. McNeil_, May 09 2010