A059193 Engel expansion of 1/e = 0.367879... .

3, 10, 28, 54, 88, 130, 180, 238, 304, 378, 460, 550, 648, 754, 868, 990, 1120, 1258, 1404, 1558, 1720, 1890, 2068, 2254, 2448, 2650, 2860, 3078, 3304, 3538, 3780, 4030, 4288, 4554, 4828, 5110, 5400, 5698, 6004, 6318, 6640, 6970, 7308, 7654, 8008, 8370, 8740

Offset

1,1

Comments

Cf. A006784 for definition of Engel expansion.

References

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

Links

G. C. Greubel and T. D. Noe, Table of n, a(n) for n = 1..1000[Terms 1 to 300 computed by T. D. Noe; Terms 301 to 1000 computed by G. C. Greubel, Dec 27 2016]

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner 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.

Peter J. Larcombe, Jack Sutton, and James Stanton, A note on the constant 1/e, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619.

Eric Weisstein's World of Mathematics, Engel Expansion

Index entries for sequences related to Engel expansions

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 2*(2*n+1)*(n-1) (for n>1) follows from 1/e = Sum_{n>=1} (1/(2*n)! - 1/(2*n+1)!). - Helena Verrill (verrill(AT)math.lsu.edu), Jan 19 2004

a(1)=3, a(2)=10, a(1)=28, a(2)=54, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 10 2012

From G. C. Greubel, Dec 27 2016: (Start)

G.f.: x*(3 + x + 7*x^2 - 3*x^3)/(1-x)^3.

E.g.f.: 2 + 3*x + 2*(2*x^2 + x - 1)*exp(x). (End)

Mathematica

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[1/E, 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)
Join[{3}, Table[2*(2*n+1)*(n-1), {n, 1, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Join[{3}, LinearRecurrence[{3, -3, 1}, {10, 28, 54}, 50]] (* Harvey P. Dale, May 10 2012 *)

Prog

(PARI) Vec(x*(3 + x + 7*x^2 - 3*x^3)/(1-x)^3 + O(x^50)) \\ G. C. Greubel, Dec 27 2016

Crossrefs

Cf. A068985.

Sequence in context: A373279 A379297 A262724 * A226863 A325793 A325800

Adjacent sequences: A059190 A059191 A059192 * A059194 A059195 A059196

Keyword

nonn,easy,nice

Author

Mitch Harris

Status

approved