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A094885 Decimal expansion of phi*e, where phi = (1 + sqrt(5))/2.

%I A094885 #18 Feb 16 2025 08:32:53
%S A094885 4,3,9,8,2,7,2,3,8,9,4,4,7,9,4,6,3,9,5,9,7,1,9,8,7,0,2,9,2,9,2,8,8,5,
%T A094885 8,6,8,7,8,6,7,4,0,4,9,7,9,7,8,8,3,4,9,1,7,0,3,8,0,9,8,0,9,0,2,1,6,4,
%U A094885 4,4,4,3,2,1,1,6,2,0,4,4,3,0,0,3,8,5,4,6,4,3,5,2,9,2,9,4,7,2,6
%N A094885 Decimal expansion of phi*e, where phi = (1 + sqrt(5))/2.
%C A094885 Matches the value of the infinite nested radical corresponding to the sequence {e^(2^n), n=1,2,3,...}, i.e., a = sqrt(e^2+sqrt(e^4+...)), which converges by Vijayaraghavan's theorem. Proof: write the golden ratio as phi = sqrt(1+ sqrt(1+ sqrt(1+...))). Then e*phi = e*sqrt(1+ sqrt(1+ sqrt(1+ ...))) = sqrt(e^2+ e^2*sqrt(1+ sqrt(1+ ...))) = sqrt(e^2+ sqrt(e^4+ e^4*sqrt(1+ ...))) = ... = a. Evidently, the 'e' could stand for any constant, not just e; for example phi itself as in A104457, or Pi as in A094886. - _Stanislav Sykora_, May 24 2016
%H A094885 Harry J. Smith, <a href="/A094885/b094885.txt">Table of n, a(n) for n = 1..20000</a>
%H A094885 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NestedRadical.html">Nested Radical</a>
%H A094885 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e A094885 4.398272389447946...
%t A094885 First@ RealDigits[N[GoldenRatio E, 120]] (* _Michael De Vlieger_, May 24 2016 *)
%o A094885 (PARI) default(realprecision, 20080); phi=(1+sqrt(5))/2; x=phi*exp(1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b094885.txt", n, " ", d)); \\ _Harry J. Smith_, Apr 27 2009
%o A094885 (PARI) exp(1)*(1+sqrt(5))/2 \\ _Michel Marcus_, May 25 2016
%Y A094885 Cf. A001113, A001622, A104457, A094886.
%K A094885 cons,nonn
%O A094885 1,1
%A A094885 _N. J. A. Sloane_, Jun 15 2004