OFFSET
1,1
COMMENTS
Cf. A006784 for definition of Engel expansion.
For n >= 2, the total number of self-crossings of a {2*n+1 / n} regular star polygon. - Charles L. Hohn, Dec 08 2025
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 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)
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)
a(n) = 2 * A014106(n-1) for n >= 2. - Charles L. Hohn, Dec 08 2025
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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved
