A160238 Numbers n such that n^2 can be expressed as the sum of three different nonzero Fibonacci numbers.

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, 123, 139, 216, 322, 843, 1161, 1476, 2207, 3864, 4999, 5778, 15127, 39603, 103682, 271443, 710647, 1244196, 1860498, 4870847, 12752043

Offset: 1

Author

Carmine Suriano, May 05 2009

Keywords

• nonn

Comments

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).

Examples

			4999*4999=24990001=Fib(37)+Fib(35)+Fib(3)
		

Crossrefs

Cf. A000045, A135709, A135558.

Extensions

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