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Search: id:A013643
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| A013643 |
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Numbers n such that continued fraction for sqrt(n) has period 3. |
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+0
1
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| 41, 130, 269, 370, 458, 697, 986, 1313, 1325, 1613, 1714, 2153, 2642, 2834, 3181, 3770, 4409, 4778, 4933, 5098, 5837, 5954, 6626, 7465, 7610, 8354, 9293, 10282, 10865, 11257, 11321, 12410, 13033, 13549, 14698, 14738, 15977, 17266, 17989
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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All numbers of the form (5n+1)^2 + 4n + 1 for n>0 are elements of this sequence. Numbers of the above form have the continued fraction expansion [5n+1,[2,2,10n+2]]. General square roots of integers with period 3 continued fraction expansions have expansions of the form [n,[2m,2m,2n]]. - David Terr (David_C_Terr(AT)raytheon.com), Jun 15 2004
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REFERENCES
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Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors in this reference!)
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FORMULA
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The general form of these numbers is d = d(m, n) = a^2 + 4mn + 1, where m and n are positive integers and a = a(m, n) = (4m^2 + 1)n + m, for which the continued fraction expansion of sqrt(d) is [a;[2m, 2m, 2a]]. - David Terr (David_C_Terr(AT)raytheon.com), Jul 20 2004
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CROSSREFS
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Cf. A044292 A044673 A067896 A028343 A044373 A044754.
Adjacent sequences: A013640 A013641 A013642 this_sequence A013644 A013645 A013646
Sequence in context: A135792 A067896 A142290 this_sequence A142333 A028343 A165816
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Clark Kimberling (ck6(AT)evansville.edu), Walter Gilbert
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