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 A057592 Fibonacci(n+1)^2+4*Fibonacci(n). 0
 1, 5, 8, 17, 37, 84, 201, 493, 1240, 3161, 8141, 21092, 54865, 143061, 373608, 976609, 2554357, 6683444, 17491097, 45781949, 119841976, 313723305, 821294493, 2150106052, 5628936097, 14736560549, 38580516296, 101004617393, 264432735685, 692292618516 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Theorem: only the first term is a square. Proof from Don Coppersmith: (F[n+1] + 2)^2 = F[n+1]^2 + 4*F[n+1] + 4 > F[n+1]^2 + 4*F[n]. But (F[n+1] + 1)^2 -(F[n+1]^2 + 4*F[n])= 2*F[n+1] + 1 - 4*F[n] is odd and positive, so can't be 0. Thus our number is trapped between 2 successive squares. REFERENCES Postings to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) by Victor S. Miller, Oct 05 2000. LINKS MATHEMATICA Table[Fibonacci[n + 1]^2 + 4*Fibonacci[n], {n, 0, 200}] (* and *) CoefficientList[Series[(-4 z^4 + 7 z^3 + 8 z^2 - 2 z - 1)/(z^5 - z^4 - 5 z^3 + z^2 + 3 z - 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 30 2011 *) CROSSREFS Cf. A000045. Sequence in context: A091625 A027601 A261808 * A246638 A192170 A280251 Adjacent sequences:  A057589 A057590 A057591 * A057593 A057594 A057595 KEYWORD nonn AUTHOR N. J. A. Sloane, Oct 05 2000 STATUS approved

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