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Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=6.
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%I #16 Oct 15 2024 15:42:43

%S 8,9,8,6,5,3,7,1,2,6,2,8,6,9,9,2,9,3,2,6,0,8,7,5,7,2,2,0,4,6,8,0,5,8,

%T 8,6,2,6,0,4,4,8,2,2,0,0,9,3,4,3,9,6,9,6,6,8,5,5,3,1,5,5,6,5,4,7,3,2,

%U 5,8,4,7,0,1,7,2,1,9,7,8,2,4,6,8,7,6,8

%N Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=6.

%C Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-6.

%H Paolo P. Lava, <a href="/A230154/b230154.txt">Table of n, a(n) for n = 0..1000</a>

%F Equals 1/A230160. - _Hugo Pfoertner_, Oct 15 2024

%e 0.8986537126286992932608757220468058862604482200934396966...

%p with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);

%p for n from q by -1 to 1 do a:=(1+a)^(1/h);od;

%p print(evalf(a,1000)); end: P(1000,-6);

%t RealDigits[x/.FindRoot[x^7+x^6==1,{x,1},WorkingPrecision->120]][[1]] (* _Harvey P. Dale_, Dec 30 2013 *)

%Y Cf. A075778, A230151, A230152, A230153, A230155, A230156, A230157, A230158, A230160, A377081.

%K nonn,cons

%O 0,1

%A _Paolo P. Lava_, Oct 11 2013