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EXAMPLE
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Write F(n) for the n-th Fibonacci number (F(0) = 0). We omit writing 0s below. Then examples of numbers that can be written as a sum of the squares of 4 Fibonacci numbers are 0 = F(0)^2, 1 = F(1)^2, 2 = F(1)^2 + F(1)^2, 3 = F(1)^2 + F(1)^2 + F(1)^2 and 4 = F(3)^2 = F(1)^2 + F(1)^2 + F(1)^2 + F(1)^2, the first number with different representations (ignoring F(1) = 1 = F(2)).
If we write (n,a,b,c,d) with a<=b<=c<=d to mean that n = F(a)^2 + F(b)^2 + F(c)^2 + F(d)^2, then an exhaustive list of representations for the integers in [5,23] is (5,0,0,1,3), (6,0,1,1,3), (7,1,1,1,3), (8,0,0,3,3), (9,0,0,0,4), (9,0,1,3,3), (10,0,0,1,4) or (10,1,1,3,3), (11,0,1,1,4), (12,0,3,3,3), (13,0,0,3,4) or (13,1,3,3,3), (14,0,1,3,4), (15,1,1,3,4), (16,3,3,3,3), (17,0,3,3,4), (18,0,0,4,4) or (18,1,3,3,4), (19,0,1,4,4), (20,1,1,4,4), (21,3,3,3,4), (22,0,3,4,4), and (23,1,3,4,4). Among all integers up to 100,000, the only two numbers to have three representations are 178 with (178,0,4,4,5), (178,0,2,2,6), and (178,0,0,3,6); and 196 with (196,2,5,5,5), (196,3,3,3,6), and (196,0,0,4,6).
In the examples of the complementary sequence A364353, it is explained why 24 and 32 have no representation.
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