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Search: id:A057592
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| A057592 |
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Fibonacci(n+1)^2+4*Fibonacci(n). |
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+0
1
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| 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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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.
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REFERENCES
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Postings to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) by Victor S. Miller, Oct 05 2000.
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CROSSREFS
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Sequence in context: A031191 A091625 A027601 this_sequence A104321 A026595 A034453
Adjacent sequences: A057589 A057590 A057591 this_sequence A057593 A057594 A057595
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KEYWORD
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nonn
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AUTHOR
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njas, Oct 05 2000
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