A230161 -id:A230161 - cp's OEIS frontend

A230160 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=7.

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

Offset: 1

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=7.

			1.1127756842787054706297040205710929356068592718552836814...

Paolo P. Lava, Table of n, a(n) for n = 1..1000 (corrected by _Sean A. Irvine_)

Cf. A001622, A060006, A060007, A160155, A230159, A230161-A230163.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,7);

Root[x^7 - x - 1, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A230162 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=9.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=9.

			1.0850702454914508283368958640973142340506536310308965814...

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A001622, A060006, A060007, A160155, A230159-A230161, A230163.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,9);

Root[(#^9-#-1)&, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[x/.FindRoot[x^9-x-1==0,{x,1},WorkingPrecision->100]][[1]] (* Harvey P. Dale, Jul 31 2017 *)

A298814 Decimal expansion of the greatest real zero of x^8 - 2*x^4 - x + 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 13 2018

Let (d(n)) = (0,0,1,0,0,1,...), s(n) = (s(n-1) + d(n))^(1/2) for n > 0, and s(0) = 1.
Then s(3n) -> 1.096981557798..., as in A230161;
s(3n+1) -> 1.2033685381501..., as in A298814;
s(3n+2) -> 1.44809583860..., as in A298815.

			Greatest real zero = 1.2033685381501...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A230161, A298813, A298815.

r = x /. NSolve[x^8 - 2 x^4 - x + 1 == 0, x, 10000][[8]];
RealDigits[r][[1]]; (* A298814 *)

solve(x=1, 2, x^8 - 2*x^4 - x + 1) \\ Michel Marcus, Nov 05 2018

A298815 Decimal expansion of the greatest real zero of x^8 - 4x^6 + 6x^4 - 4*x^2 - x + 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 13 2018

Let (d(n)) = (0,0,1,0,0,1,...), s(n) = (s(n-1) + d(n))^(1/2) for n > 0, and s(0) = 1.
Then s(3n) -> 1.096981557798... , as in A230161;
s(3n+1) -> 1.2033685381501..., as in A298814;
s(3n+2) -> 1.44809583860..., as in A298815.

			greatest real zero = 1.4480958386096...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A230161, A298813, A298814.

r = x /. NSolve[x^8 - 4 x^6 + 6 x^4 - 4 x^2 - x  + 1 == 0, x, 10000][[8]];
RealDigits[r][[1]]; (* A298815 *)

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A230160 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=7.

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A230162 Decimal expansion of the positive real solution of the equation x^k-x-1=0. Case k=9.

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A298814 Decimal expansion of the greatest real zero of x^8 - 2*x^4 - x + 1.

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A298815 Decimal expansion of the greatest real zero of x^8 - 4*x^6 + 6*x^4 - 4*x^2 - x + 1.

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A298815 Decimal expansion of the greatest real zero of x^8 - 4x^6 + 6x^4 - 4*x^2 - x + 1.