A320428 -id:A320428 - cp's OEIS frontend

A160510 Decimal expansion of exp(Pi/4).

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

Offset: 1

Hagen von Eitzen, May 16 2009

Identified by Knuth as one of those "quantities that are frequently used in standard subroutines and in analysis of computer programs." - Alonso del Arte, Feb 03 2012

			Exp(Pi/4) = 2.1932800507380154565597696592787382234616+ according to Knuth, appendix B, table 1.

D. E. Knuth, The Art Of Computer Programming, Vol 1: Fundamental Algorithms, Addison-Wesley, 1968.

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

Cf. A000796, A320428 (continued fraction), A329912 (Engel expansion).

evalf(exp(Pi/4), 125);  # Alois P. Heinz, Nov 17 2019

RealDigits[ E^(Pi/4), 10, 111][[1]] (* Robert G. Wilson v, May 29 2009 *)

exp(Pi/4) \\ Charles R Greathouse IV, Jan 04 2016

More terms from Robert G. Wilson v, May 29 2009

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);

cp's OEIS Frontend

A160510 Decimal expansion of exp(Pi/4).

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