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 A059193 Engel expansion of 1/e = 0.367879... . 3
 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 (list; graph; refs; listen; history; text; internal format)
 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. ErdÅ‘s 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. 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 ((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: A127912 A005956 A262724 * A226863 A325793 A325800 Adjacent sequences:  A059190 A059191 A059192 * A059194 A059195 A059196 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified October 18 15:21 EDT 2019. Contains 328162 sequences. (Running on oeis4.)