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A263719 Decimal expansion of the real root r of r^3 + r - 1 = 0.

%I A263719 #33 Jan 20 2024 09:17:49
%S A263719 6,8,2,3,2,7,8,0,3,8,2,8,0,1,9,3,2,7,3,6,9,4,8,3,7,3,9,7,1,1,0,4,8,2,
%T A263719 5,6,8,9,1,1,8,8,5,8,1,8,9,7,9,9,8,5,7,7,8,0,3,7,2,8,6,0,6,6,3,9,8,9,
%U A263719 6,6,7,8,6,8,6,9,9,8,0,2,1,0,8,1,7,3,2,0,4,3,7,8,6,2,0,5,1,2,8,2,9,5,5,9,3,3,1,8,7,6
%N A263719 Decimal expansion of the real root r of r^3 + r - 1 = 0.
%C A263719 Constant from Narayana's cows sequence: Limit A000930(n)/A000930(n+1) = r.
%C A263719 Reciprocal of constant described by A092526.
%H A263719 G. C. Greubel, <a href="/A263719/b263719.txt">Table of n, a(n) for n = 0..5000</a>
%H A263719 Cyril Banderier and Michael Wallner, <a href="https://arxiv.org/abs/1707.01931">Lattice paths with catastrophes</a>, arXiv:1707.01931 [math.CO], 2017. See Corollary 4.22. on p. 24.
%H A263719 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F A263719 r = (sqrt(93)/18 + 1/2)^(1/3) - (sqrt(93)/18 - 1/2)^(1/3).
%F A263719 Constant r satisfies:
%F A263719 (1) 1/(1 - r*i) = (r + r^2*i) where i^2 = -1.
%F A263719 (2) r = real( 1/(1 - r*i) ).
%F A263719 (3) r = norm( 1/(1 - r*i) ).
%F A263719 (4) r = r^2 + r^4.
%F A263719 Equals 1/A092526. - _Vaclav Kotesovec_, Nov 27 2017
%e A263719 0.682327803828019327369483739711048256891188581897998577803728606639896...
%t A263719 RealDigits[ ((Sqrt[93] + 9)/18)^(1/3) - ((Sqrt[93] - 9)/18)^(1/3), 10, 100][[1]] (* _G. C. Greubel_, May 01 2017 *)
%o A263719 (PARI) a(n) = my(r = (sqrt(93)/18 + 1/2)^(1/3) - (sqrt(93)/18 - 1/2)^(1/3)); floor(r*10^(n+1))%10
%o A263719 for(n=0,120,print1(a(n),", "))
%o A263719 (PARI) solve(r=0, 1,  r^3 + r - 1 ) \\ _Michel Marcus_, Oct 25 2015
%Y A263719 Cf. A000930, A025157, A076725, A092526, A274112.
%K A263719 nonn,cons
%O A263719 0,1
%A A263719 _Paul D. Hanna_, Oct 24 2015