A059193 -id:A059193 - cp's OEIS frontend

A068985 Decimal expansion of 1/e.

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

Offset: 0

N. J. A. Sloane, Apr 08 2002

From the "derangements" problem: this is the probability that if a large number of people are given their hats at random, nobody gets their own hat.
Also, decimal expansion of cosh(1)-sinh(1). - Mohammad K. Azarian, Aug 15 2006
Also, this is lim_{n->inf} P(n), where P(n) is the probability that a random rooted forest on [n] is a tree. See linked file. - Washington Bomfim, Nov 01 2010
Also, location of the minimum of x^x. - Stanislav Sykora, May 18 2012
Also, -1/e is the global minimum of x*log(x) at x = 1/e and the global minimum of x*e^x at x = -1. - Rick L. Shepherd, Jan 11 2014
Also, the asymptotic probability of success in the secretary problem (also known as the sultan's dowry problem). - Andrey Zabolotskiy, Sep 14 2019
The asymptotic density of numbers with an odd number of trailing zeros in their factorial base representation (A232745). - Amiram Eldar, Feb 26 2021
For large range size s where numbers are chosen randomly r times, the probability when r = s that a number is randomly chosen exactly 1 time. Also the chance that a number was not chosen at all. The general case for the probability of being chosen n times is (r/s)^n / (n! * e^(r/s)). - Mark Andreas, Oct 25 2022

			1/e = 0.3678794411714423215955237701614608674458111310317678... = A135005/5.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Sections 1.3 and 5,23,3, pp. 14, 409.
Anders Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).
John Harris, Jeffry L. Hirst, and Michael Mossinghoff, Combinatorics and Graph Theory, Springer Science & Business Media, 2009, p. 161.
L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (103) on page 20.
Traian Lalescu, Problem 579, Gazeta Matematică, Vol. 6 (1900-1901), p. 148.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
Manfred R. Schroeder, Number Theory in Science and Communication, Springer Science & Business Media, 2008, ch. 9.5 Derangements.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 26, page 233.
Walter D. Wallis and John C. George, Introduction to Combinatorics, CRC Press, 2nd ed. 2016, theorem 5.2 (The Derangement Series).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Washington Bomfim, Probabilities of connected random forests and derangements, Oct 31 2010.
James Grime and Brady Haran, Derangements, Numberphile video, 2017.
Peter J. Larcombe, Jack Sutton, and James Stanton, A note on the constant 1/e, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619.
Gérard P. Michon, Final Answers: Inclusion-Exclusion.
Michael Penn, A cool, quick limit, YouTube video, 2022.
Eric Weisstein's World of Mathematics, Derangement.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Spherical Bessel Function of the First Kind.
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem.
Eric Weisstein's World of Mathematics, e.
OEIS Wiki, Number of derangements.
Index entries for transcendental numbers.

Cf. A000166, A001113, A068996, A092553, A232745.
Cf. A059193.
Cf. asymptotic probabilities of success for other "nothing but the best" variants of the secretary problem: A325905, A242674, A246665.
Cf. A049470, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

RealDigits[N[1/E,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

default(realprecision, 110);
exp(-1) \\ Rick L. Shepherd, Jan 11 2014

Equals 2*(1/3! + 2/5! + 3/7! + ...). [Jolley]
Equals 1 - Sum_{i >= 1} (-1)^(i - 1)/i!. [Michon]
Equals lim_{x->infinity} (1 - 1/x)^x. - Arkadiusz Wesolowski, Feb 17 2012
Equals j_1(i)/i = cos(i) + i*sin(i), where j_1(z) is the spherical Bessel function of the first kind and i = sqrt(-1). - Stanislav Sykora, Jan 11 2017
Equals Sum_{i>=0} ((-1)^i)/i!. - Maciej Kaniewski, Sep 10 2017
Equals Sum_{i>=0} ((-1)^i)(i^2+1)/i!. - Maciej Kaniewski, Sep 12 2017
From Peter Bala, Oct 23 2019: (Start)
The series representation 1/e = Sum_{k >= 0} (-1)^k/k! is the case n = 0 of the following series acceleration formulas:
1/e = n!*Sum_{k >= 0} (-1)^k/(k!*R(n,k)*R(n,k+1)), n = 0,1,2,..., where R(n,x) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*k!*binomial(-x,k) are the row polynomials of A094816. (End)
1/e = 1 - Sum_{n >= 0} n!/(A(n)*A(n+1)), where A(n) = A000522(n). - Peter Bala, Nov 13 2019
Equals Integral_{x=0..1} x * sinh(x) dx. - Amiram Eldar, Aug 14 2020
Equals lim_{x->oo} (x!)^(1/x)/x. - L. Joris Perrenet, Dec 08 2020
Equals lim_{n->oo} (n+1)!^(1/(n+1)) - n!^(1/n) (Lalescu, 1900-1901). - Amiram Eldar, Mar 29 2022

More terms from Rick L. Shepherd, Jan 11 2014

A159200 Decimal expansion of Sum_{k >= 1} (1/(10^(4k + 2) - 1)) - (1/(10^(2k + 1) - 1)), negated.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 3, 0, 3, 0, 1, 0, 3, 0, 1, 0, 1, 0, 5, 0, 1, 0, 2, 0, 3, 0, 1, 0, 3, 0, 3, 0, 1, 0, 1, 0, 5, 0, 3, 0, 1, 0, 3, 0, 1, 0, 1, 0, 5, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 3, 0, 1, 0, 3, 0, 3, 0, 3, 0, 1, 0, 5, 0

Offset: 0

Eric Desbiaux, Apr 06 2009

It equals Sum_{k >= 1} 1/((2^(4*k + 2)*5^(4*k + 2)) - 1) - 1/((2^(2*k + 1)*5^(2*k + 1)) - 1).
Note that Sum_{k >= 1} (1/(10^k - 1)) / Sum_{k >= 1} ((1/(10^(4*k + 2) - 1)) -(1/(10^(2*k + 1) - 1))) = A073668 / Sum_{k >= 1} ((1/(10^(4*k + 2) - 1)) - (1/(10^(2*k + 1) - 1))) = -121.100.
My idea for this decimal expansion came from the Engel expansion of e - 1, i.e., A000027(n) = n, and the Engel expansion of e^(-1), i.e., A059193(n) = 2*(2*n + 1)*(n - 1), which I have transformed into (2*n + 1)^2 - (6*n + 3) (since 2*(2*n + 1)*(n - 1) = (2*n + 1)^2 - (6*n + 3)). It appears that the Engel expansion of 1/e works like a Sundaram sieve.
Decimal expansion of Sum_{n>=0} (d(2*n+1) - 1)/(10^(2*n+1) - 1), where d = A000005. - Jianing Song, Apr 12 2021

			-0.00101010201010301010301020301010303010301010501020301030301...

Eric Weisstein's World of Mathematics, Engel expansion.
English Wikipedia, Sieve of Sundaram.
French Wikipedia, Crible de Sundaram.

suminf(k=1, 1/(10^(4*k + 2) - 1) - 1/(10^(2*k + 1) - 1)) \\ Michel Marcus, Jun 25 2019

Comments edited by Petros Hadjicostas, Jun 19 2019

A329451 Maximum number of pieces that can be captured during one move on an n X n board according to the international draughts capture rules.

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 5, 9, 10, 16, 19, 25, 28, 36, 41, 49, 54, 64, 71, 81, 88, 100, 109, 121, 130, 144, 155, 169, 180, 196, 209, 225, 238, 256, 271, 289, 304, 324, 341, 361, 378, 400, 419, 441, 460, 484, 505, 529, 550, 576, 599, 625, 648, 676, 701, 729, 754, 784

Offset: 0

Stéphane Rézel, Nov 14 2019

nonn

Captures are made diagonally, forward and backward. Kings have the long-range capturing capability. During the multiple capture, a piece may pass over the same empty square several times, but no opposing piece can be jumped twice. Captured pieces can only be lifted from the board after the end of the multiple capture.

			It is possible to capture in a single move 19 opposing pieces on a 10 X 10 board, but not one more, so a(10) = 19.

Stéphane Rézel, Table of n, a(n) for n = 0..1000
Fabien Gigante, Solution to Problem J122 La grande rafle de la dame, Diophante, 2009 (in French).
Stéphane Rézel, Solution to Problem J128 La méga-rafle de la dame, Diophante, 2020 (in French).
World Draughts Federation, Official rules for international draughts.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A000290, A059193, A125202, A000982 (active squares).

a(n) = if(n<5, floor(n/3), (n^2 - 2*n + if(n%2, 1, 2*(n%4) - 8))/4)

a(2*t+1) = t^2 = A000290(t).
a(4*t+6) = 4*t^2 + 10*t + 5 = A125202(t+2).
a(4*t+8) = 4*t^2 + 14*t + 10 = A059193(t+2).
a(0) = a(2) = 0; a(4) = 1.
Recurrence: For t >= 1, a(2*t+1) = a(2*t-1) + 2*t - 1;
For t >= 1, a(4*t+3) = a(4*t+2) + 2*t + 2; a(4*t+2) = a(4*t+1) + 2*t - 1;
For t >= 2, a(4*t+1) = a(4*t) + 2*t + 2; a(4*t) = a(4*t-1) + 2*t - 3.
From Colin Barker, Nov 14 2019: (Start)
G.f.: x^3*(1 - x + 3*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + 2*x^6 - x^7) / ((1 - x)^3*(1 + x)*(1 + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>10.
(End)

A364823 Triangle read by rows: T(n,k) = number of possible positions for four connected discs in the game "Connect Four" played on a board with n columns and k rows, 4 <= k <= n.

Original entry on oeis.org

10, 17, 28, 24, 39, 54, 31, 50, 69, 88, 38, 61, 84, 107, 130, 45, 72, 99, 126, 153, 180, 52, 83, 114, 145, 176, 207, 238, 59, 94, 129, 164, 199, 234, 269, 304, 66, 105, 144, 183, 222, 261, 300, 339, 378, 73, 116, 159, 202, 245, 288, 331, 374, 417, 460

Offset: 4

Felix Huber, Aug 09 2023

In the game, all these positions can be reached. The most difficult thing is to connect four discs in the top row in the case of n=k. Here are examples for 4 X 4, 5 X 5 and 6 X 6:
                                             .  b3 b12  b8 b11  .
                      b3  b5  b8 b10  .      .  a3 a12  b7 a11  .
  b2  b4  b8  b7      b2  a5  a8 a10  .      .  b2 b10  a7 a10  .
  a2  a4  a8  b6      a2  b4  b7  b9  .      .  a2  a8  b6  b9  .
  b1  b3  a7  a6      b1  a4  a7  a9  .      .  b1  a6  b5  a9  .
  a1  a3  b5  a5      a1  a3  b6  a6  .      .  a1  b4  a4  a5  .
For n >= 7 any position in the top row can be reached by the following procedure. By repeating the following scheme, a tower of any height up to the second highest row can be built by placing discs alternately:
  b4  b3  a4  a3
  a1  a2  b1  b2
You can also build a separate tower where you are completely free with at least three discs. While one player places his four discs in the top row, the other moves to these reserve squares. Therefore, any position of four connected discs in the top row can be realized. Example 7 X 7:
  .   a   a   a   a   .   .
  .   b   b   a   a   .   .
  .   a   a   b   b   .   .
  .   b   b   a   a   .   .
  .   a   a   b   b   .   b
  .   b   b   a   a   .   b
  .   a   a   b   b   .   b
For vertical positions there are many reserve squares in the other columns, for diagonal and horizontal positions other than in the top row you have additional reserve squares above three of the four discs to connect. For n > k you have further columns with more reserve squares.

			The triangle T(n,k) begins:
  n/k   4     5     6     7     8     9    10 ...
   4:  10
   5:  17    28
   6:  24    39    54
   7:  31    50    69    88
   8:  38    61    84   107   130
   9:  45    72    99   126   153   180
  10:  52    83   114   145   176   207   238
   .
   .
   .

Wikipedia, Connect Four.

Cf. A013582, A090224, A212693, A059193.

A364823 := proc(n) local k; for k from 4 to n do return 4*k*n - 9*k - 9*n + 18; end do; end proc; seq(A364823(n), n = 4 .. 100);

T(n,k) = 4*k*n - 9*k - 9*n + 18, 4 <= k <= n, comprising k*(n-3) = k*n - 3*k horizontal positions, n*(k-3) = k*n - 3*n vertical positions, and 2*(n-3)*(k-3) = 2*k*n - 6*k - 6*n + 18 diagonal positions.

T(n,n) = 4*n^2 - 18*n + 18 = A059193(n-2).

cp's OEIS Frontend

A068985 Decimal expansion of 1/e.

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A159200 Decimal expansion of Sum_{k >= 1} (1/(10^(4*k + 2) - 1)) - (1/(10^(2*k + 1) - 1)), negated.

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A329451 Maximum number of pieces that can be captured during one move on an n X n board according to the international draughts capture rules.

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PARI

Formula

A364823 Triangle read by rows: T(n,k) = number of possible positions for four connected discs in the game "Connect Four" played on a board with n columns and k rows, 4 <= k <= n.

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A159200 Decimal expansion of Sum_{k >= 1} (1/(10^(4k + 2) - 1)) - (1/(10^(2k + 1) - 1)), negated.