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A190178 Continued fraction of (1+sqrt(2)+sqrt(7+6*sqrt(2)))/2.

%I A190178 #14 Jun 25 2025 01:08:33
%S A190178 3,5,1,2,1,1,1,2,1,12,1,5,1,1,2,1,14,2,9,11,1,12,1,2,1,832,1,2,2,5,1,
%T A190178 1,17,1,2,1,9,1,12,1,1,1,6,3,2,1,1,6,3,1,1,1,2,2,1,3,1,3,3,1,2,1,45,1,
%U A190178 1,1,1,62,9,1,1,2,3,1,6,1,3,5,1,4
%N A190178 Continued fraction of (1+sqrt(2)+sqrt(7+6*sqrt(2)))/2.
%C A190178 Equivalent to the periodic continued fraction [r,1,r,1,...] where r=1+sqrt(2), the silver ratio. For geometric interpretations of both continued fractions, see A189977 and A188635.
%H A190178 G. C. Greubel, <a href="/A190178/b190178.txt">Table of n, a(n) for n = 1..10000</a>
%t A190178 r = 1 + 2^(1/2);
%t A190178 FromContinuedFraction[{r, 1, {r, 1}}]
%t A190178 FullSimplify[%]
%t A190178 ContinuedFraction[%, 100]  (* A190178 *)
%t A190178 RealDigits[N[%%, 120]]     (* A190177 *)
%t A190178 N[%%%, 40]
%t A190178 ContinuedFraction[(1 + Sqrt[2] + Sqrt[7 + 6*Sqrt[2]])/2, 100] (* _G. C. Greubel_, Dec 28 2017 *)
%o A190178 (PARI) contfrac((1+sqrt(2)+sqrt(7+6*sqrt(2)))/2) \\ _G. C. Greubel_, Dec 28 2017
%o A190178 (Magma) ContinuedFraction((1+Sqrt(2)+Sqrt(7+6*Sqrt(2)))/2); // _G. C. Greubel_, Dec 28 2017
%Y A190178 Cf. A188635, A190177, A190180.
%K A190178 nonn,cofr
%O A190178 1,1
%A A190178 _Clark Kimberling_, May 05 2011