A248788 -id:A248788 - cp's OEIS frontend

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

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).

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).

A248789 Decimal expansion of the variance associated with the fraction of guests without a napkin in Conway’s napkin problem.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 14 2014

			0.034763105561026065633697454779470105240123600705...

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

Cf. A000670, A068996, A248788.

Join[{0}, RealDigits[(3 - E)*(2 - Sqrt[E])^2, 10, 99] // First]

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

Equals (3 - e)*(2 - sqrt(e))^2.

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Python

Python

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Python

Formula

A248789 Decimal expansion of the variance associated with the fraction of guests without a napkin in Conway’s napkin problem.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula