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

Author

Mitch Harris

Keywords

• nonn
• easy
• nice

Comments

Cf. A006784 for definition of Engel expansion.

References

• Friedrich 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]
• Friedrich 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.
• Paul 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).

Crossrefs

Cf. A006784, A068985.

Programs

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 *)

PARI

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

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)

From Amiram Eldar, May 05 2025: (Start)

Sum_{n>=1} 1/a(n) = 7/9 - log(2)/3.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1/9 + Pi/12 - log(2)/6. (End)