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A090550 Decimal expansion of solution to n/x = x - n for n = 5.

%I A090550 #31 Sep 02 2024 11:03:26
%S A090550 5,8,5,4,1,0,1,9,6,6,2,4,9,6,8,4,5,4,4,6,1,3,7,6,0,5,0,3,0,9,6,9,1,4,
%T A090550 3,5,3,1,6,0,9,2,7,5,3,9,4,1,7,2,8,8,5,8,6,4,0,6,3,4,5,8,6,8,1,1,5,7,
%U A090550 8,1,3,8,8,4,5,6,7,0,7,3,4,9,1,2,1,6,2,1,6,1,2,5,6,8,1,7,3,4,1,2,4
%N A090550 Decimal expansion of solution to n/x = x - n for n = 5.
%C A090550 n/x = x - n with n = 1 gives the Golden Ratio = 1.6180339887...
%C A090550 Equals n + n/(n + n/(n + n/(n + ....))) for n = 5.  See also A090388. - _Stanislav Sykora_, Jan 23 2014
%H A090550 Chai Wah Wu, <a href="/A090550/b090550.txt">Table of n, a(n) for n = 1..10001</a>
%H A090550 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A090550 n/x = x - n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 5: x = (5 + sqrt(45))/2 = 5.85410196624968454...
%F A090550 Equals (5 + 3*sqrt(5))/2 = 1 + 3*phi = sqrt(5)*(phi)^2, where phi is the golden ratio. - _G. C. Greubel_, Jul 03 2017
%F A090550 Equals 2*phi^3 - phi^2. - _Michel Marcus_, Apr 20 2020
%F A090550 Minimal polynomial is x^2 - 5x - 5 (this number is an algebraic integer). - _Alonso del Arte_, Apr 20 2020(n).
%F A090550 Equals lim_{n->oo} A057088(n+1)/A057088(n) = 1 + 3*phi. - _Wolfdieter Lang_, Nov 16 2023
%e A090550 5.85410196624968454...
%t A090550 RealDigits[(5 + 3 Sqrt[5])/2, 10, 120][[1]] (* _Harvey P. Dale_, Nov 27 2013 *)
%o A090550 (PARI) (5 + 3*sqrt(5))/2 \\ _G. C. Greubel_, Jul 03 2017
%Y A090550 Cf. n + n/(n + n/(n + ...)): A090388 (n = 2), A090458 (n = 3), A090488 (n = 4), A092294 (n = 6), A092290 (n = 7), A090654 (n = 8), A090655 (n = 9), A090656 (n = 10). - _Stanislav Sykora_, Jan 23 2014
%Y A090550 Cf. A001622, A057088.
%K A090550 easy,nonn,cons
%O A090550 1,1
%A A090550 _Felix Tubiana_, Feb 05 2004