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Search: id:A098316
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| A098316 |
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Decimal expansion of [3, 3, ...] = (3 + Sqrt[13])/2. |
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+0
8
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| 3, 3, 0, 2, 7, 7, 5, 6, 3, 7, 7, 3, 1, 9, 9, 4, 6, 4, 6, 5, 5, 9, 6, 1, 0, 6, 3, 3, 7, 3, 5, 2, 4, 7, 9, 7, 3, 1, 2, 5, 6, 4, 8, 2, 8, 6, 9, 2, 2, 6, 2, 3, 1, 0, 6, 3, 5, 5, 2, 2, 6, 5, 2, 8, 1, 1, 3, 5, 8, 3, 4, 7, 4, 1, 4, 6, 5, 0, 5, 2, 2, 2, 6, 0, 2, 3, 0, 9, 5, 4, 1, 0, 0, 9, 2, 4, 5, 3, 5, 8, 8, 3
(list; cons; graph; listen)
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OFFSET
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1,1
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COMMENT
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If c is this constant and n > 0, then for n even, c^n = [A100230(n), 1, A100230(n)-1, 1, A100230(n)-1, 1, A100230(n)-1, 1, ...], for n odd, c^n = [A100230(n)+1, A100230(n)+1, A100230(n)+1, ...]. - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Dec 15 2007
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FORMULA
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3 plus the constant in A085550. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 02 2008]
Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 02 2009 (Start): Set c:=(3+sqrt(13))/2. Then the fractional part of c^n equals 1/c^n, if n odd. For even n, the fractional part of c^n is equal to 1-(1/c^n).
c:=(3+sqrt(13))/2 suffices c-c^(-1)=floor(c)=3, hence c^n+(-c)^(-n)=nint(c^n) for n>0, which follows from the general formula of A001622.
1/c=(sqrt(13)-3)/2.
See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which suffice x-x^(-1)=floor(x).
Other examples of constants x satisfying the relation x-x^(-1)=floor(x) include A001622 (the golden ratio: where floor(x)=1) and A014176 (the silver ratio: where floor(x)=2).
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EXAMPLE
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3.30277563...
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CROSSREFS
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Cf. A001622, A014176, A098317, A098318.
Cf. A000032, A006497, A080039.
Sequence in context: A141947 A010607 A118522 this_sequence A160165 A084055 A084103
Adjacent sequences: A098313 A098314 A098315 this_sequence A098317 A098318 A098319
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KEYWORD
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nonn,cons,easy
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AUTHOR
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Eric Weisstein (eric(AT)weisstein.com), Sep 02, 2004
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EXTENSIONS
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For reasons following from the formula section, this constant could be called "the bronze ratio". For this, compare with A001622 and A014176. (End)
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