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A059193
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Engel expansion of 1/e = 0.367879... .
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5
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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
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OFFSET
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1,1
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COMMENTS
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Cf. A006784 for definition of Engel expansion.
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REFERENCES
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F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.
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LINKS
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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.
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FORMULA
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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
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)
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MATHEMATICA
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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}, LinearRecurrence[{3, -3, 1}, {10, 28, 54}, 50]] (* Harvey P. Dale, May 10 2012 *)
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PROG
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(PARI) Vec(x*(3 + x + 7*x^2 - 3*x^3)/(1-x)^3 + O(x^50)) \\ G. C. Greubel, Dec 27 2016
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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