Antoine Beaulieu - cp's OEIS frontend

A358630 Decimal expansion of a seed to the logistic map with r=4 such that mapping the orbit to 0 and 1 gives the binary expansion of Pi.

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

Offset: 0

Antoine Beaulieu, Nov 24 2022

This constant, when fed as seed to the logistic map f(x) = 4*x(1-x), gives an orbit that maps to the binary expansion of Pi with 1 if x_i > 0.5 and 0 if x_i < 0.5.

This seed to 300 decimal digits is good to generate about 1000 binary digits of Pi.

			x_0 = 0.585730... -> 1
x_1 = 0.970601... -> 1
x_2 = 0.114139... -> 0
x_3 = 0.404444... -> 0
x_4 = 0.963477... -> 1

Cf. A004601.

A350417 a(n) is the maximum number of rows of width n that can be filled by stacking squares of sizes 1 through n without creating voids or overhangs.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 12, 12, 16, 15, 20, 24, 26, 25, 30, 29, 35, 33, 40, 40, 47, 46, 61, 51, 73, 56, 63, 61, 71, 71, 82, 82, 90, 83, 96, 95, 108, 99, 108, 111, 127, 102, 146, 144, 130, 150, 150, 161, 144, 160, 175, 173, 186, 162, 183, 179, 200, 182, 202, 197, 226

Offset: 1

Antoine Beaulieu, Dec 29 2021

nonn

No square of any size can be used more than once. The initial terms listed are conjectured from what I can draw. Trying to stack squares in this way feels like playing Tetris and comes from the squared square idea (see A295130).

			|‾‾‾‾‾‾‾‾‾‾‾|
|           |
|     5     |
|           |        a(7)=12
|___________||‾‾‾|
|‾‾‾‾‾‾‾‾||_||___|
|    4   ||‾‾‾‾‾‾|
|        ||  3   |
|________||______|
|       7        |
.
.
               |‾‾‾‾‾‾|
|‾‾‾‾‾‾‾‾‾‾‾‾‾||   3  |      a(10)=15
|             ||      ||‾|
|       6     ||‾‾‾‾‾‾‾‾‾|
|             ||    4    |
|             ||         |
|_____________||_________|
|           10           |

Pontus von Brömssen, A350417(61) = 226.

Cf. A295130.

a(19) corrected by, a(21) and beyond from Pontus von Brömssen, Nov 15 2022

A328011 The 5x + 1 sequence beginning at 1.

Original entry on oeis.org

1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1, 6, 3, 16, 8, 4, 2, 1

Offset: 0

Antoine Beaulieu, Oct 01 2019

See A328010 for further information.

Colin Barker, Table of n, a(n) for n = 0..1000
Alex V. Kontorovich & Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems, arXiv:0910.1944 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1).

Cf. A259207, A328010.

Vec((1 + 6*x + 3*x^2 + 16*x^3 + 8*x^4 + 4*x^5 + 2*x^6) / ((1 - x)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^80)) \\ Colin Barker, Oct 08 2019

a(n+1) = 5*a(n) + 1 if a(n) is odd, a(n+1) = a(n)/2 otherwise.
From Colin Barker, Oct 08 2019: (Start)
G.f.: (1 + 6*x + 3*x^2 + 16*x^3 + 8*x^4 + 4*x^5 + 2*x^6) / ((1 - x)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = a(n-7) for n>6.
(End)

A328010 The 5x + 1 sequence beginning at 17.

Original entry on oeis.org

17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17

Offset: 0

Antoine Beaulieu, Oct 01 2019

The 5x+1 problem is similar to the 3x+1 or Collatz problem. For some starting values it is known that the 5x+1 trajectory will tend to infinity or enter a periodic orbit.
Alex V. Kontorovich & Jeffrey C. Lagarias conjectured that there are very few periodic orbits. One of them is shown here.
The two other known periodic orbits are given in the crossrefs.

Colin Barker, Table of n, a(n) for n = 0..1000
Alex V. Kontorovich & Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems arXiv:0910.1944 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A259207, A328011.

Vec((17 + 86*x + 43*x^2 + 216*x^3 + 108*x^4 + 54*x^5 + 27*x^6 + 136*x^7 + 68*x^8 + 34*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Oct 05 2019

a(n+1) = 5*a(n) + 1 if a(n) is odd, a(n+1) = a(n)/2 otherwise.
From Colin Barker, Oct 04 2019: (Start)
G.f.: (17 + 86*x + 43*x^2 + 216*x^3 + 108*x^4 + 54*x^5 + 27*x^6 + 136*x^7 + 68*x^8 + 34*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-10) for n>9.
(End)

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User: Antoine Beaulieu

A358630 Decimal expansion of a seed to the logistic map with r=4 such that mapping the orbit to 0 and 1 gives the binary expansion of Pi.

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A350417 a(n) is the maximum number of rows of width n that can be filled by stacking squares of sizes 1 through n without creating voids or overhangs.

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A328011 The 5x + 1 sequence beginning at 1.

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A328010 The 5x + 1 sequence beginning at 17.

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