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A124132
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Numbers n such that F_{2n} = a^2 + b^2, where F_m is the m-th Fibonacci number and a, b are integers. Note that this excludes all Fibonacci numbers with odd indices, which all have this property.
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0
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1, 3, 6, 7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 111, 127, 163, 169, 183, 199
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Fibonacci numbers with even indices factor as F_2n=F_n L_n, L_n being n-th Lucas number. Thus F_2n factors further as F_2n = F_m L_m L_2m L_4m...L_n, with m odd. Since F_m is a sum of two squares and the pairs of Lucas numbers all have GCD dividing 2, the conclusion for F_2n depends on each Lucas number being a sum of two squares. Joint work with Kevin O'Bryant and Dennis Eichhorn.
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EXAMPLE
| a(4)=7 because the first four Fibonacci numbers with even indices that are the sum of two squares are F_2, F_6, F_12 and F_14, 14 being 2*a(4) and F_14=377=11^2+16^2.
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CROSSREFS
| Cf. A000032, A000204 = Lucas numbers, A000045 = Fibonacci numbers.
Sequence in context: A107850 A051218 A124130 * A064291 A137473 A127307
Adjacent sequences: A124129 A124130 A124131 * A124133 A124134 A124135
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KEYWORD
| nonn
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AUTHOR
| Melvin J. Knight (melknightdr(AT)verizon.net), Nov 30 2006
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