1,1

Eric Weisstein's World of Mathematics, Silver Ratio

Wikipedia, Silver ratio

Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 02 2009 (Start): Set c:=1+sqrt(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:=1+sqrt(2) satisfies c-c^(-1)=floor(c)=2, hence c^n+(-c)^(-n)=nint(c^n) for n>0, which follows from the general formula of A001622.

1/c=sqrt(2)-1.

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 A098316 (the "bronze" ratio: where floor(x)=3). (End)

Cf. A002193.

Cf. A000032, A006497, A080039.

Sequence in context: A157284 A143973 A011167 this_sequence A060047 A135185 A011029

Adjacent sequences: A014173 A014174 A014175 this_sequence A014177 A014178 A014179

nonn,cons

N. J. A. Sloane (njas(AT)research.att.com).