A341232 -id:A341232 - cp's OEIS frontend

A248788 Decimal expansion of (2-sqrt(e))^2, the mean fraction of guests without a napkin in Conway’s napkin problem.

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

Offset: 0

Jean-François Alcover, Oct 14 2014

			0.12339674565853264796568432009600821114214269085936752866665...

Anders Claesson and T. Kyle Petersen, Conway's napkin problem, arXiv:math/0505080 [math.CO], 2005.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 2.

Cf. A000670, A068996, A248789, A341232, A341233.

RealDigits[(2 - Sqrt[E])^2, 10, 100] // First

(2-exp(1/2))^2 \\ Charles R Greathouse IV, Oct 31 2014

Equals lim_{n->oo} A341232(n)/A341233(n). - Pontus von Brömssen, Feb 08 2021

A341233 Denominator of the expected fraction of guests without a napkin in Conway's napkin problem with n guests.

Original entry on oeis.org

1, 1, 12, 96, 320, 3840, 161280, 516096, 46448640, 185794560, 2270822400, 163499212800, 1821848371200, 51011754393600, 10712468422656000, 9794256843571200, 555007887802368000, 139861987726196736000, 1449478781889675264000, 49059281848573624320000

Offset: 1

Pontus von Brömssen, Feb 07 2021

			0, 0, 1/12, 11/96, 39/320, 473/3840, 19897/161280, 63683/516096, 5731597/46448640

Cf. A248788, A341232 (numerators).

from sympy import denom, S, factorial
def A341233(n):
  return denom(sum((1-S(2)**(2-k))/factorial(k) for k in range(2,n+1)))

from math import factorial
from fractions import Fraction
def a(n):
  s = sum(Fraction(2**k-4, 2**k*factorial(k)) for k in range(2, n+1))
  return s.denominator
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Feb 07 2021

A341232(n)/a(n) = Sum_{k=2..n} (1-2^(2-k))/k!.

Lim_{n->oo} A341232(n)/a(n) = (2-sqrt(e))^2 (A248788).

cp's OEIS Frontend

A248788 Decimal expansion of (2-sqrt(e))^2, the mean fraction of guests without a napkin in Conway’s napkin problem.

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