A346192 -id:A346192 - cp's OEIS frontend

A087046 Algebraic order of r_n, the value of r in the logistic map that corresponding to the onset of the period 2^n-cycle.

1, 2, 12, 240, 65280, 4294901760, 18446744069414584320, 340282366920938463444927863358058659840, 115792089237316195423570985008687907852929702298719625575994209400481361428480

Offset: 1

Eric W. Weisstein, Aug 04 2003

Ilias S. Kotsireas and Kostas Karamanos, Exact computation of the bifurcation point B4 of the logistic map and the Bailey-Broadhurst conjectures, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Vol. 14, No. 7 (2004), pp. 2417-2423; alternative link.
Eric Weisstein's World of Mathematics, Logistic Map.

Cf. A086180, A086181, A087089, A118454, A000215, A346192.

Cf. A051179 (partial sums).

Table[If[n <= 1, 1, 2^(2^(n - 1)) - 2^(2^(n - 2))], {n, 1, 10}] (* Cheng Zhang, Apr 02 2012 *)

a(n) = 1<<(1<<(n-1)) - 1<<(1<<(n-2)); \\ Kevin Ryde, Jan 18 2024

a(n) = 2^(2^(n - 1)) - 2^(2^(n - 2)) with n>1, a(1)=1. - Cheng Zhang, Apr 02 2012

Sum_{n>=1} 1/a(n) = 1 + A346192. - Amiram Eldar, Jul 18 2021

More terms from Cheng Zhang, Apr 03 2012

A346190 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k+1)) - 1).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 09 2021

This constant is transcendental (Schwarz, 1967).

			0.50781250046566128730773925781250000000587747175411...

Wolfgang Schwarz, Remarks on the irrationality and transcendence of certain series, Mathematica Scandinavica, Vol. 20, No. 2 (1967), pp. 269-274; alternative link.
Index entries for transcendental numbers

Cf. A048649, A076214, A346191, A346192.

RealDigits[Sum[1/(2^(2^(2*n + 1) - 1)), {n, 0, 10}], 10, 100][[1]]

Equals A076214 - A346191.

A346191 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k)) - 1).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 09 2021

This constant is transcendental (Schwarz, 1967).

			1.12503051757812500010842021724855044340074528008699...

Wolfgang Schwarz, Remarks on the irrationality and transcendence of certain series, Mathematica Scandinavica, Vol. 20, No. 2 (1967), pp. 269-274; alternative link.
Index entries for transcendental numbers

Cf. A048649, A076214, A346190, A346192.

RealDigits[Sum[1/(2^(2^(2*n) - 1)), {n, 0, 10}], 10, 100][[1]]

Equals A076214 - A346190.

cp's OEIS Frontend

A087046 Algebraic order of r_n, the value of r in the logistic map that corresponding to the onset of the period 2^n-cycle.

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A346190 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k+1)) - 1).

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A346191 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k)) - 1).

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