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%I A318605 #64 Dec 14 2024 07:17:55
%S A318605 2,8,9,0,0,5,3,6,3,8,2,6,3,9,6,3,8,1,2,4,5,7,0,0,9,2,9,6,1,0,3,1,2,9,
%T A318605 6,0,9,4,3,5,9,1,7,2,2,1,6,4,5,8,5,9,1,1,0,7,5,2,0,8,9,0,0,5,2,4,4,5,
%U A318605 5,8,0,3,8,3,5,4,9,7,0,4,6,1,5,3,7,5,9,1,4,1,9,1,7,7,8,5,1,3,9,6,0,2,3,2,6,8
%N A318605 Decimal expansion of geometric progression constant for Coxeter's Loxodromic Sequence of Tangent Circles.
%C A318605 This constant and its reciprocal are the real solutions of x^4 - 2*x^3 - 2*x^2 - 2*x + 1 = (x^2 - (sqrt(5)+1)*x + 1)*(x^2 + (sqrt(5)-1)*x + 1) = 0.
%C A318605 This constant and its reciprocal are the solutions of x^2 - (1+sqrt(5))*x + 1 = 0.
%C A318605 Decimal expansion of the largest x satisfying x^2 - (1+sqrt(5))*x + 1 = 0.
%C A318605 For sequences of type aa(n) = 2*(aa(n-1) + aa(n-2) + aa(n-3)) - aa(n-4) for arbitrary initial terms (except the trivial all zero), i.e., linear recurrence relations of order 4 with signature (2,2,2,-1), lim_{n -> infinity} aa(n)/aa(n-1) = this constant; see for instance A192234, A192237, A317973, A317974, A317975, A317976.
%H A318605 A.H.M. Smeets, <a href="/A318605/b318605.txt">Table of n, a(n) for n = 1..9999</a> (terms 1..3000 from Muniru A Asiru)
%H A318605 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F A318605 Equals A001622 + A139339, i.e., phi + sqrt(phi) where phi is the golden ratio.
%e A318605 2.8900536382639638124570092961031296094359...
%p A318605 evalf[180]((1+sqrt(5))/2+sqrt((1+sqrt(5))/2)); # _Muniru A Asiru_, Nov 21 2018
%t A318605 RealDigits[GoldenRatio + Sqrt[GoldenRatio], 10 , 120][[1]] (* _Amiram Eldar_, Nov 22 2018 *)
%o A318605 (PARI) ((1+sqrt(5))/2 + sqrt((1+sqrt(5))/2)) \\ _Michel Marcus_, Nov 21 2018
%Y A318605 Cf. A001622, A139339, A192234, A192237, A317973, A317974, A317975, A317976.
%K A318605 nonn,cons
%O A318605 1,1
%A A318605 _A.H.M. Smeets_, Sep 07 2018

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