Search: id:A098316
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%I A098316
%S A098316 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,
%T A098316 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,
%U A098316 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
%N A098316 Decimal expansion of [3, 3, ...] = (3 + Sqrt[13])/2.
%C A098316 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
%F A098316 3 plus the constant in A085550. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Sep 02 2008]
%F A098316 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).
%F A098316 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.
%F A098316 1/c=(sqrt(13)-3)/2.
%F A098316 See A001622 for a general formula concerning the fractional parts of 
               powers of numbers x>1, which suffice x-x^(-1)=floor(x).
%F A098316 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).
%e A098316 3.30277563...
%Y A098316 Cf. A001622, A014176, A098317, A098318.
%Y A098316 Cf. A000032, A006497, A080039.
%Y A098316 Sequence in context: A141947 A010607 A118522 this_sequence A160165 A084055 
               A084103
%Y A098316 Adjacent sequences: A098313 A098314 A098315 this_sequence A098317 A098318 
               A098319
%K A098316 nonn,cons,easy
%O A098316 1,1
%A A098316 Eric Weisstein (eric(AT)weisstein.com), Sep 02, 2004
%E A098316 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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