Philipp O. Tsvetkov - cp's OEIS frontend

A307154 Decimal expansion of the fraction of occupied places on an infinite lattice cover with 3-length segments.

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

Offset: 0

Philipp O. Tsvetkov, Mar 27 2019

Solution of the discrete parking problem when infinite lattice randomly filled with 3-length segments.
Solution of the discrete parking problem when infinite lattice randomly filled with 2-length segments is equal to 1-1/e^2 (see A219863).
Also, the limit of a(n) = (3 + 2*(n-3)*a(n-3) + (n-1)*(n-3)*a(n-1))/(n*(n-2)); a(0) = 0; a(1) = 0; a(2) = 0 as n tends to infinity.
If the length of the segments that randomly cover infinite lattice tends to infinity, then the fraction of occupied places is equal to Rényi's parking constant (see A050996).

			0.8236529631773383369006718778116478872139236620539298680914372350071822...

D. G. Radcliffe, Fat men sitting at a bar
Philipp O. Tsvetkov, Stoichiometry of irreversible ligand binding to a one-dimensional lattice, Scientific Reports, Springer Nature (2020) Vol. 10, Article number: 21308.
Eric Weisstein's World of Mathematics, Dawson's Integral

Cf. A050996, A087654, A099288, A219863, A307131, A307132.

evalf(3*sqrt(Pi)*(erfi(2)-erfi(1))/(2*exp(4)), 120) # Vaclav Kotesovec, Mar 28 2019

N[-((3 DawsonF[1])/E^3) + 3 DawsonF[2], 200] // RealDigits

-imag(3*sqrt(Pi)*(erfc(2*I) - erfc(1*I)) / (2*exp(4))) \\ Michel Marcus, May 10 2019

Equals 3*(Dawson(2) - Dawson(1)/e^3).

Equals 3*sqrt(Pi)*(erfi(2) - erfi(1)) / (2*exp(4)).

A307132 Denominator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

Original entry on oeis.org

1, 3, 6, 5, 45, 63, 420, 405, 14175, 17325, 187110, 552825, 14189175, 49116375, 729729000, 723647925, 8881133625, 109185701625, 2062396586250, 10257709336875, 428772250281375, 2348038513445625, 53791427762572500, 160789593855515625, 16025362854266390625

Offset: 1

Philipp O. Tsvetkov, Mar 26 2019

The limit of expected fraction of occupied places on n-length lattice randomly filled with 2-length segments at n tends to infinity is equal to 1-1/e^2 (see A219863).

			0, 1, 2/3, 5/6, 4/5, 37/45, 52/63, 349/420, 338/405, 11873/14175, ...

D. G. Radcliffe, Fat men sitting at a bar

Cf. A219863, A231580, A307131 (numerators).

RecurrenceTable[{f[n] == (2 + 2 (n - 2) f[n - 2] + (n - 1) (n - 2) f[n - 1])/(n (n - 1)), f[0] == 0, f[1] == 0}, f, {n, 2, 100}] // Denominator

Denominator of f(n), where f(0)=0; f(1)=0 and f(n) = (2 + 2(n-2)f(n-2) + (n-1)(n-2)f(n-1))/(n(n-1)) for n>1.

A307131 Numerator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

Original entry on oeis.org

1, 2, 5, 4, 37, 52, 349, 338, 11873, 14554, 157567, 466498, 11994551, 41582906, 618626159, 614191052, 7545655031, 92853583996, 1755370057489, 8737266957604, 365468962351379, 2002633668589496, 45904893141293831

Offset: 1

Philipp O. Tsvetkov, Mar 26 2019

The limit of expected fraction of occupied places on n-length lattice randomly filled with 2-length segments at n tends to infinity is equal to 1-1/e^2 (see A219863).

			0, 1, 2/3, 5/6, 4/5, 37/45, 52/63, 349/420, 338/405, 11873/14175, ...

D. G. Radcliffe, Fat men sitting at a bar

Cf. A219863, A231580, A307132 (denominators).

RecurrenceTable[{f[n] == (2 + 2 (n - 2) f[n - 2] + (n - 1) (n - 2) f[n - 1])/(n (n - 1)),f[0] == 0, f[1] == 0}, f, {n, 2, 100}] // Numerator

Numerator of f(n), where f(0)=0; f(1)=0 and f(n) = (2 + 2(n-2)f(n-2) + (n-1)(n-2)f(n-1))/(n(n-1)) for n>1.

A307053 Decimal expansion of e + Pi + ePi + e^Pi + Pi^e + Pisqrt(e) + e*sqrt(Pi).

Original entry on oeis.org

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

Offset: 2

Philipp O. Tsvetkov, Mar 21 2019

Curiously, the values of A105643 and A307054 are close to the integers 60 and 10, respectively, which leads to the value of this constant being close to 70. - Felix Fröhlich, May 26 2019

			69.99709878241019348543095089938317296397...

Cf. A105643, A307054.

RealDigits[N[Pi + Pi^E + E Pi + E^Pi + E + E Sqrt[Pi] + Pi Sqrt[E], 170]][[1]]

my(e=exp(1)); e + Pi + e*Pi + e^Pi + Pi^e + Pi*sqrt(e) + e*sqrt(Pi) \\ Michel Marcus, Mar 27 2019

Equals A307054 + A105643.

A307184 Decimal expansion of the fraction of occupied places on an infinite lattice cover with 4-length segments.

Original entry on oeis.org

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

Offset: 0

Philipp O. Tsvetkov, Mar 28 2019

The solution of the discrete parking problem when infinite lattice randomly filled with L-length segments at L=4.
At L=3 it is equal to 3*(Dawson(2) - Dawson(1)/e^3) (see A307154).
At L=2 it is equal to 1-1/e^2 (see A219863).
The general solution of the discrete parking problem when infinite lattice randomly filled with L-length segments is equal to L*e(-2H(L-1))*Integral_{x=0..1} e^(2*(t + t^2/2 + t^3/3 + ... + t^(L-1)/(L-1))) dx, where H(L) is harmonic number.
Also, the limit of the following recurrence as n tends to infinity: a(n) = (4 + 2(n-4)*a(n-4) + (n-1)*(n-4)*a(n-1))/(n*(n-3)); a(0) = 0; a(1) = 0; a(2) = 0; a(3) = 0.
If L tends to infinity, then the fraction of occupied places is equal to Rényi's parking constant (see A050996).

			0.80389347991537697266629741950321342054687916485770835923972993280709456095...

D. G. Radcliffe, Fat men sitting at a bar
Philipp O. Tsvetkov, Stoichiometry of irreversible ligand binding to a one-dimensional lattice, Scientific Reports, Springer Nature (2020) Vol. 10, Article number: 21308.

Cf. A219863, A050996, A307132, A307131, A307154.

evalf(Integrate(4*exp(2*(t + t^2/2 + t^3/3) - 11/3), t= 0..1), 120); # Vaclav Kotesovec, Mar 28 2019

RealDigits[ N[(4*Integrate[E^(2*(t + t^2/2 + t^3/3)), {t, 0, 1}])/E^(11/3), 200]][[1]]

intnum(t=0, 1, 4*exp(2*(t + t^2/2 + t^3/3) - 11/3)) \\ Michel Marcus, May 10 2019

4*Integral_{x=0..1} e^(2*(t + t^2/2 + t^3/3)) dx / e^(11/3).

A306883 Decimal expansion of the minimum value of the function f(x) = x^x^x^x.

Original entry on oeis.org

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

Offset: 0

Philipp O. Tsvetkov, Mar 15 2019

The minimum value of the function x^x^x^x which is obtained for x equal to A306579.

			0.5932372977697284645520601979470817...

Cf. A306579.

n = 500;
(FindMinimum[x^x^x^x, {x, 0.34}, WorkingPrecision -> 3 n][[1]] // RealDigits)[[1]][[;; n]]

my(y = solve(x=0.1, 1, 1 + x^x*log(x)*(1 + x*log(x)*(1 + log(x))))); y^y^y^y \\ Michel Marcus, Mar 27 2019

A307054 Decimal expansion of Pisqrt(e) + esqrt(Pi).

Original entry on oeis.org

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

Offset: 1

Philipp O. Tsvetkov, Mar 21 2019

			9.997639726547473466988230602712150679126422...

Cf. A105643, A307053.

RealDigits[N[E Sqrt[Pi] + Pi Sqrt[E], 170]][[1]]

A306579 Decimal expansion of the real number x such that f(x) = x^x^x^x is a minimum.

Original entry on oeis.org

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

Offset: 0

Philipp O. Tsvetkov, Mar 15 2019

It satisfies 1 + x^x*log(x)*(1 + x*log(x)*(1 + log(x))) = 0.

The function x^x has a minimum at x = 1/e (A068985).

			0.274689385297063462413625300538145857578998865431277705412818636280...

Cf. A068985, A306883.

n = 500;
(x /.  FindMinimum[x^x^x^x, {x, 0.34}, WorkingPrecision -> 3 n][[2]][[1]] // RealDigits)[[1]][[;; n]]

solve(x=0.1, 1, 1 + x^x*log(x)*(1 + x*log(x)*(1 + log(x)))) \\ Michel Marcus, Mar 15 2019

A319606 a(n) is that generation of the rule-30 1D cellular automaton started from a single ON cell in which n successive OFF cells appears for the first time after a(n-1).

Original entry on oeis.org

1, 4, 5, 9, 11, 21, 34, 45, 51, 88, 106, 131, 137, 158, 193, 251, 517, 772, 1029, 1283, 1539, 1794, 2052, 2305, 2561, 4101, 5121, 8197, 10241, 12291, 16388, 20482, 32772, 36865, 49154, 57345, 65539, 262150, 294913, 786437, 851969, 1310724, 1441793, 1835011

Offset: 0

Philipp O. Tsvetkov, Sep 24 2018

nonn

OFF cells outside the triangle of active cells are ignored.

			The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle:
1                          .
2                        . . .
3                      . . 0 0 .
4                    . . 0 . . . .
5                  . . 0 0 . 0 0 0 .
6                . . 0 . . . . 0 . . .
7              . . 0 0 . 0 0 0 0 . 0 0 .
8            . . 0 . . . . 0 0 . . . . . .
9          . . 0 0 . 0 0 0 . . . 0 0 0 0 0 .
10       . . 0 . . . . 0 . . 0 0 . 0 0 0 . . .
11     . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 .
12   . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . .
13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 .
0 OFF cell appears for the first time in generation (line) 1, thus a(0) = 1;
1 consecutive OFF cells (0) appear for the first time after line 1 in generation (line) 4, thus a(1) = 4;
2 consecutive OFF cells (00) appear for the first time after (line) 4 in generation (line) 5, thus a(2) = 5. [Corrected by _Rémy Sigrist_, Jul 06 2020]

Rémy Sigrist, C program for A319606

Cf. A317530.

C
```
See Links section.
```

CellularAutomaton[30, {{1}, 0}, 20000];
(Reverse[Internal`DeleteTrailingZeros[
      Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
ls = Table[
   Max[Differences[Position[Flatten@{1, %[[n]], 1}, 1]]] - 1, {n, 1,
    20000}];
res = {1};
Table[Position[ls, n] // Flatten, {n, 100}];
For[n = 1, n < 40, n++,
AppendTo[res, (Select[%[[n]], # > Last[res] &, 1][[1]])]]
res

Data corrected and more terms from Rémy Sigrist, Jul 06 2020

A319780 a(n) is the period of cyclic structures that appear in the 3-state (0,1,2) 1D cellular automaton started from a single cell at state 1 with rule n.

Original entry on oeis.org

2, 2, 1, 0, 2, 1, 0, 2, 1, 2, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 0, 1, 0, 0, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 0, 2, 1, 0, 2, 1

Offset: 1

Philipp O. Tsvetkov, Sep 27 2018

The length of the sequence is equal to 3^3^3 = 7625597484987.

			1D cellular automaton with rule=1 gives the following generations:
   1  ..........1.......... <------ start
   2  111111111...111111111 <------ end
   3  ..........1..........
   4  111111111...111111111
   5  ..........1..........
   6  111111111...111111111
   7  ..........1..........
The period is 2, thus a(1) = 2.
For rule=150:
   1  ..........1..... <------ start
   2  .........22..... <------ end
   3  ........1.......
   4  .......22.......
   5  ......1.........
   6  .....22.........
   7  ....1...........
The period is 2, thus a(150) = 2.
For rule=100000000797:
   1  .........1....... <------ start
   2  ........2.2......
   3  ........111......
   4  .......2.112.....
   5  .......12........
   6  ......21.........
   7  ........2........ <------ end
   8  ........1........
   9  .......2.2.......
  10  .......111.......
  11  ......2.112......
  12  ......12.........
  13  .....21..........
  14  .......2.........
  15  .......1.........
The period is 7, thus a(100000000797) = 7.
a(10032729) = 12.
a(10096524) = 16.

Cf. A180001.

Table[
  Length[
  Last[
   FindTransientRepeat[(Internal`DeleteTrailingZeros[
        Reverse[Internal`DeleteTrailingZeros[#]]]) & /@
     CellularAutomaton[{i, 3}, {ConstantArray[0, 25], {1}, ConstantArray[0, 25]} // Flatten, 50], 2]]],
{i, 1, 1000}
]

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User: Philipp O. Tsvetkov

A307154 Decimal expansion of the fraction of occupied places on an infinite lattice cover with 3-length segments.

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A307132 Denominator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

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A307131 Numerator of the expected fraction of occupied places on n-length lattice randomly filled with 2-length segments.

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A307053 Decimal expansion of e + Pi + e*Pi + e^Pi + Pi^e + Pi*sqrt(e) + e*sqrt(Pi).

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A307184 Decimal expansion of the fraction of occupied places on an infinite lattice cover with 4-length segments.

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A306883 Decimal expansion of the minimum value of the function f(x) = x^x^x^x.

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A307054 Decimal expansion of Pi*sqrt(e) + e*sqrt(Pi).

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A306579 Decimal expansion of the real number x such that f(x) = x^x^x^x is a minimum.

A307053 Decimal expansion of e + Pi + ePi + e^Pi + Pi^e + Pisqrt(e) + e*sqrt(Pi).

A307054 Decimal expansion of Pisqrt(e) + esqrt(Pi).