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%I #47 Sep 25 2023 14:37:51
%S 3,10,28,54,88,130,180,238,304,378,460,550,648,754,868,990,1120,1258,
%T 1404,1558,1720,1890,2068,2254,2448,2650,2860,3078,3304,3538,3780,
%U 4030,4288,4554,4828,5110,5400,5698,6004,6318,6640,6970,7308,7654,8008,8370,8740
%N Engel expansion of 1/e = 0.367879... .
%C Cf. A006784 for definition of Engel expansion.
%D F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.
%H G. C. Greubel and T. D. Noe, <a href="/A059193/b059193.txt">Table of n, a(n) for n = 1..1000</a>[Terms 1 to 300 computed by T. D. Noe; Terms 301 to 1000 computed by G. C. Greubel, Dec 27 2016]
%H F. Engel, <a href="/A006784/a006784.pdf">Entwicklung der Zahlen nach Stammbruechen</a>, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
%H P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
%H Peter J. Larcombe, Jack Sutton, and James Stanton, <a href="https://pjm.ppu.edu/sites/default/files/papers/PJM_12%282%29_2023_609_to_619.pdf">A note on the constant 1/e</a>, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F 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
%F 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
%F From _G. C. Greubel_, Dec 27 2016: (Start)
%F G.f.: x*(3 + x + 7*x^2 - 3*x^3)/(1-x)^3.
%F E.g.f.: 2 + 3*x + 2*2*x^2 + x - 1)*exp(x). (End)
%t EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
%t NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
%t EngelExp[N[1/E, 7!], 100] (* Modified by _G. C. Greubel_, Dec 27 2016 *)
%t Join[{3}, Table[2*(2*n+1)*(n-1), {n, 1, 200}]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2011 *)
%t Join[{3},LinearRecurrence[{3,-3,1},{10,28,54},50]] (* _Harvey P. Dale_, May 10 2012 *)
%o (PARI) Vec(x*(3 + x + 7*x^2 - 3*x^3)/(1-x)^3 + O(x^50)) \\ _G. C. Greubel_, Dec 27 2016
%Y Cf. A068985.
%K nonn,easy,nice
%O 1,1
%A _Mitch Harris_
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