A229139 Smallest m such that Fibonacci(2n-1) = m^2 + k^2.

0, 1, 1, 2, 3, 5, 8, 9, 21, 34, 55, 89, 73, 13, 377, 610, 987, 64, 244, 4155, 4554, 10946, 2191, 28657, 15857, 74957, 34022, 29811, 50481, 134104, 832040, 162589, 387938, 711703, 1556305, 6229800, 4173137, 4059539, 1972951, 51797450, 4866315, 165580141, 46049477, 202620393, 348451533, 181781990

Offset

1,4

Comments

Every odd-indexed Fibonacci number (A000045) is a sum of two squares (see A124134).

Which of the a(n) are not Fibonacci numbers?

Links

Table of n, a(n) for n=1..46.

Example

A000045(2*6-1) = 89 = 5^2 + 8^2 so a(6)=5.
A000045(2*8-1) = 610 = 9^2 + 23^2 = 13^2 + 21^2, so a(8)=9.

Prog

(PARI) for(n=1, 10^6, t=fibonacci(2*n-1); s=sqrtint(t); forstep(i=s, 1, -1, if(issquare(t-i*i), print1(sqrtint(t-i*i), ", "); break)))
(Haskell)
a229139 1 = 0
a229139 n = head $
   dropWhile (== 0) $ map (a037213 . (t -) . (^ 2)) [s, s - 1 ..]
   where t = a000045 (2 * n - 1); s = a000196 t
-- Reinhard Zumkeller, Oct 11 2013

Crossrefs

Cf. A000045.

Cf. A000196, A037213.

Sequence in context: A099422 A294913 A056903 * A293277 A331864 A272669

Adjacent sequences: A229136 A229137 A229138 * A229140 A229141 A229142

Keyword

nonn

Author

Ralf Stephan, Sep 15 2013

Status

approved