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%I #4 Mar 30 2012 17:39:59
%S 3,4,5,6,7,8,9,10,12,16,17,18,20,23,24,25,32,33,35,37,40,47,57,86,112,
%T 123,139,216,322,843,1161,1476,2207,3864,4999,5778,15127,39603,103682,
%U 271443,710647,1244196,1860498,4870847,12752043
%N Numbers n such that n^2 can be expressed as the sum of three different nonzero Fibonacci numbers.
%C There exist a proper subsequence b(i)of a(n): n=[1, 2, 8, 17, 21, 24, 25, 28,29, 30, 31, 32, 33, 34, ...] such that approximatively b(i+1)=b(i)*(1+phi) where phi is 1.618... is the golden ratio and the approximation holds as a limit when i goes to infinity. For such a subsequence b(i) we have the following formula for the corresponding term when squared b(i)*b(i)=Fib(4*i+1)+Fib(4*i-1)+Fib(3). In the previous example 4999=b(9).
%e 4999*4999=24990001=Fib(37)+Fib(35)+Fib(3)
%Y Cf. A000045, A135709, A135558.
%K nonn
%O 1,1
%A _Carmine Suriano_, May 05 2009
%E Inserted 4 (with 4^2=13+1+2), 6 (with 36=21+2+13), 12 (with 12^2=89+21+34) etc. Added "nonzero" to definition - _R. J. Mathar_, Oct 23 2010
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