A341233 -id:A341233 - 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

A341232 Numerator of the expected fraction of guests without a napkin in Conway's napkin problem with n guests.

Original entry on oeis.org

0, 0, 1, 11, 39, 473, 19897, 63683, 5731597, 22926439, 280212089, 20175270749, 224810160067, 6294684482461, 1321883741325001, 1208579420640469, 68486167169628137, 17258514126746312369, 178860964586279976467, 6053755724458706915971, 3305350625554453976644453

Offset: 1

Pontus von Brömssen, Feb 07 2021

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

Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters, 2004, p. 122.

Anders Claesson and T. Kyle Petersen, Conway’s napkin problem, arXiv:math/0505080 [math.CO], 2005.
Anders Claesson and T. Kyle Petersen, Conway's napkin problem, Amer. Math. Monthly 114 (No. 3, 2007), 217-231.
Niklas Eriksen, The freshman's approach to Conway's napkin problem, Amer. Math. Monthly 115 (No. 6, 2008), 492-498.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 2.

Cf. A248788, A341233 (denominators).

from sympy import numer, S, factorial
def A341232(n):
  return numer(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.numerator
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 07 2021

a(n)/A341233(n) = Sum_{k=2..n} (1-2^(2-k))/k!.

Lim_{n->oo} a(n)/A341233(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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