A000038 -id:A000038 - cp's OEIS frontend

A364896 Decimal expansion of the 4-volume of the unit regular 120-cell.

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

Offset: 3

Jianing Song, Aug 12 2023

Decimal expansion of (1575+705*sqrt(5))/4.

			Equals 787.85698103433793399211...

Polytope Wiki, Hecatonicosachoron
Wikipedia, 120-cell

Decimal expansion of 4-volumes: A364895 (5-cell), A000007 = 1 (8-cell or tesseract), A020793 = 1/6 (16-cell), A000038 = 2 (24-cell), this sequence (120-cell), A364897 (600-cell).

Cf. A102769 (decimal expansion of the volume of the unit regular dodecahedron).

First[RealDigits[(1575 + 705*Sqrt[5])/4, 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

PARI
```
(1575+705*sqrt(5))/4
```

A364897 Decimal expansion of the 4-volume of the unit regular 600-cell.

Original entry on oeis.org

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

Offset: 2

Jianing Song, Aug 12 2023

Decimal expansion of (50+25*sqrt(5))/4.

			Equals 26.47542485937368560255...

Wikipedia, 600-cell
Polytope Wiki, Hexacosichoron

Decimal expansion of 4-volumes: A364895 (5-cell), A000007 = 1 (8-cell or tesseract), A020793 = 1/6 (16-cell), A000038 = 2 (24-cell), A364896 (120-cell), this sequence (600-cell).

Cf. A102208 (decimal expansion of the volume of the unit regular icosahedron).

First[RealDigits[(50 + 25*Sqrt[5])/4, 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

PARI
```
(50+25*sqrt(5))/4
```

A337301 Triangle read by rows in which row n lists the closest integers to diagonal lengths of regular n-gon with unit edge length, n >= 4.

Original entry on oeis.org

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

Offset: 4

Mohammed Yaseen, Aug 22 2020

			Triangle begins:
1;
2, 2;
2, 2, 2;
2, 2, 2, 2;
2, 2, 3, 2, 2;
2, 3, 3, 3, 3, 2;
2, 3, 3, 3, 3, 3, 2;
2, 3, 3, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 4, 3, 3, 2;
2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2;
...
Row n lists the closest integers to the length of the diagonals drawn from a fixed vertex of a regular n-gon with unit edge length, n >= 4.
The lengths of the diagonals drawn from vertex A of a regular 8-gon ABCDEFGH with unit edge length are:
AC = 1.84775...
AD = 2.41421...
AE = 2.61312...
AF = 2.41421...
AG = 1.84775...
So the row for n=8 is 2, 2, 3, 2, 2.

Cf. A064313.
Decimal expansion of diagonal lengths of regular n-gons with unit edge length:
n=4  A002193.
n=5  A001622.
n=6  A002194, A000038.
n=7  A160389, A231187.
n=8  A179260, A014176, A121601.
n=9  A332437.
n=10 A188593, A104457, A019970, A134945.
n=11 A231186.
n=12 A188887, A090388, A337402, A019973, A214726.

T[n_,k_]:=Round[Sin[(k+1)*Pi/n]/Sin[Pi/n]]; Flatten[Table[T[n,k],{n,4,16},{k,1,n-3}]] (* Stefano Spezia, Sep 07 2020 *)

T(n,k) = round(sin((k+1)*Pi/n)/sin(Pi/n)), n >= 4, 1 <= k <= n-3.

A305187 Decimal expansion of the solution to x^x^x = 3.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, May 27 2018

Let x(m) be the solution to the equation x^x^x^...^x = m, where x appears m times on the left hand side; e.g.,
                                                decimal
    m         equation        solution x(m)    expansion
  ====  ====================  =============  =============
     1              x = 1     1.00000000...     A000007
     2            x^x = 2     1.55961046...     A030798
     3          x^x^x = 3     1.63507847...  this sequence
     4        x^x^x^x = 4     1.62036995...
     5      x^x^x^x^x = 5     1.59340881...
     6    x^x^x^x^x^x = 6     1.56864406...
     7  x^x^x^x^x^x^x = 7     1.54828598...
.
    10  x^x^x^x^...^x = 10    1.50849792...
.
   100  x^x^x^x^...^x = 100   1.44567285...
.
  1000  x^x^x^x^...^x = 1000  1.44467831...
.
Then x(1) < x(m) < x(3) for all m >= 4.
Let y(k/2) be the solution to the equation y^y^y^...^y = (k/2)*y^y, where y appears k times on the left hand side; e.g.,
                                                decimal
k          equation           solution y(k/2)  expansion
=  =========================  ===============  =========
1              y = (1/2)*y^y  2                 A000038
2            y^y = (2/2)*y^y  indeterminate
3          y^y^y = (3/2)*y^y  1.6998419085...
4        y^y^y^y = (4/2)*y^y  1.6396207046...
5      y^y^y^y^y = (5/2)*y^y  1.5987769216...
6    y^y^y^y^y^y = (6/2)*y^y  1.5694666408...
7  y^y^y^y^y^y^y = (7/2)*y^y  1.5476452822...
.
What is lim_{k -> infinity} y(k/2)?
Lim_{m -> infinity} x(m) = e^(1/e). - Jon E. Schoenfield, Jul 23 2018
Lim_{k -> infinity} y(k/2) = e^(1/e). - Jon E. Schoenfield, Aug 01 2018

			1.635078474636375245899757198787500888...

Cf. A000007, A000038, A030798.

RealDigits[ FindRoot[ x^x^x == 3, {x, 1}, WorkingPrecision -> 128][[1, 2]]][[1]] (* Robert G. Wilson v, Jun 13 2018 *)

default(realprecision,333);
solve(x=1.6, 1.7, x^x^x-3) \\ Joerg Arndt, May 27 2018

More digits from Michel Marcus, Joerg Arndt, May 27 2018

A320029 Decimal expansion of sqrt(9 + sqrt(9 + sqrt(9 + sqrt(9 + ...)))) = (sqrt(37) + 1)/2.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Oct 03 2018

For x >= 0, sqrt(x + sqrt(x + sqrt(x + sqrt(x + ...)))) = (sqrt(4*x+1) + 1)/2. This is an integer for each x such that 2*x is a term in A000217.

			3.541381265149109844499842122601033531042485047393205593209576523243166362659...

Index entries for algebraic numbers, degree 2

Cf. A000007, A001622, A000038, A209927, A222132, A222134, A223140, A235162.

evalf((sqrt(37)+1)/2,120); # Muniru A Asiru, Oct 07 2018

RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[9, {135}]], 10, 111][[1]] (* or *)
RealDigits[(Sqrt[37] + 1)/2, 10, 111][[1]]

(sqrt(37)+1)/2 \\ Altug Alkan, Oct 03 2018

Minimal polynomial: x^2 - x - 9. - Stefano Spezia, Jul 02 2025

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A364896 Decimal expansion of the 4-volume of the unit regular 120-cell.

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A364897 Decimal expansion of the 4-volume of the unit regular 600-cell.

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A337301 Triangle read by rows in which row n lists the closest integers to diagonal lengths of regular n-gon with unit edge length, n >= 4.

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A305187 Decimal expansion of the solution to x^x^x = 3.

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A320029 Decimal expansion of sqrt(9 + sqrt(9 + sqrt(9 + sqrt(9 + ...)))) = (sqrt(37) + 1)/2.

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