A160510 -id:A160510 - cp's OEIS frontend

A094390 Beatty sequence of exp(Pi/4).

2, 4, 6, 8, 10, 13, 15, 17, 19, 21, 24, 26, 28, 30, 32, 35, 37, 39, 41, 43, 46, 48, 50, 52, 54, 57, 59, 61, 63, 65, 67, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 92, 94, 96, 98, 100, 103, 105, 107, 109, 111, 114, 116, 118, 120, 122, 125, 127, 129, 131, 133, 135, 138, 140

Offset: 1

Robert G. Wilson v, Apr 28 2004

nonn

Beatty complement is A094391.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A160510, A094391.

R:= RealField(100);
[Floor(n*Exp(Pi(R)/4)): n in [1..100]]; // G. C. Greubel, Sep 27 2024

Mathematica
```
c = E^(Pi/4); Table[Floor[n*c], {n,65}]
```

[int(n*exp(pi/4)) for n in range(1,101)] # G. C. Greubel, Sep 27 2024

a(n) = floor(n * exp(Pi/4)).

A320428 Continued fraction expansion of exp(Pi/4).

Original entry on oeis.org

2, 5, 5, 1, 3, 25, 1, 1, 17, 1, 3, 3, 1, 12, 1, 8, 5, 3, 1, 46, 3, 4, 12, 1, 5, 22, 3, 2, 1, 7, 4, 2, 1, 13, 13, 8, 1, 1, 3, 1, 1, 1, 2, 1, 11, 1, 5, 2, 1, 4, 7, 1, 71, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 6, 1, 9, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 2, 1, 2, 10, 1, 19, 2, 2, 4, 1

Offset: 0

Grant T Sanderson, Aug 28 2019

This value arises naturally by taking the ratio of the volume of a unit 2n-dimensional ball to the volume of the 2n-dimensional cube containing it (with side length 2) and summing over all n.

Greg Egan, Puzzle in which this value arises naturally
Grant Sanderson and Brady Haran, Darts in Higher Dimensions, Numberphile video (2019)

Cf. A160510 (decimal expansion), A058287, A087299, A329912 (Engel expansion).

Mathematica
```
ContinuedFraction[Exp[Pi/4], 100]
```

contfrac(exp(Pi/4)) \\ Felix Fröhlich, Aug 28 2019

A329912 Engel expansion of exp(Pi/4).

Original entry on oeis.org

1, 1, 6, 7, 9, 17, 57, 283, 326, 791, 10332, 17303, 24977, 85451, 96025, 192273, 337177, 700071, 1394732, 2514757, 73904827, 176943055, 340834596, 663816066, 833303392, 2045234708, 2352089677, 7248164506, 10106625539, 32495772149, 54837573240, 60139816999

Offset: 1

Alois P. Heinz, Nov 23 2019

nonn

Cf. A006784 for definition of Engel expansion.

Greg Egan, Puzzle in which this value arises naturally
F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Grant Sanderson and Brady Haran, Darts in Higher Dimensions, Numberphile video (2019)
Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A006784, A160510 (decimal expansion), A320428 (continued fraction).

Digits:= 250:
engel:= (r, n)-> `if`(n=0 or r=0, NULL,
        [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
engel(evalf(exp(Pi/4)), 32);

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A094390 Beatty sequence of exp(Pi/4).

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A320428 Continued fraction expansion of exp(Pi/4).

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PARI

A329912 Engel expansion of exp(Pi/4).

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