A068379 Engel expansion of sinh(1/2).
2, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648, 25920, 27224, 28560
Offset: 1
Examples
sinh(1/2) = 1/2 + 1/(2*24) + 1/(2*24*80) + 1/(2*24*80*168) + 1/(2*24*80*168*288) + ... = 0.52109530549374736162242562641... = A334367.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..10001
- Eric W. Weisstein's World of Mathematics, Engel Expansion.
- Wikipedia, Engel Expansion.
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Programs
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Magma
[2] cat [8*(n*(2*n-3)+1): n in [2..50]]; // Vincenzo Librandi, Jan 31 2012
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Mathematica
Table[If[n==1, 2, 8*(n*(2*n-3)+1)], {n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *) LinearRecurrence[{3,-3,1},{2,24,80,168},50] (* Harvey P. Dale, Mar 21 2017 *) -
PARI
a(n)=if(n<=1,2,8*(n*(2*n-3)+1)) \\ Charles R Greathouse IV, Jan 31 2012
Formula
a(n) = 8*(n*(2*n-3)+1) for n > 1, a(1)=2.
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: 2*x*(1+9*x+7*x^2-x^3)/(1-x)^3. (End)
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = (3-log(2))/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/4 - Pi/16 - log(2)/8. (End)
From Elmo R. Oliveira, May 29 2025: (Start)
E.g.f.: 2*(4*exp(x)*(1 - x + 2*x^2) + (x - 4)).
a(n) = 2*A033586(n-1) for n >= 2.
a(n) = 4*A002943(n-1) for n >= 2. (End)
Extensions
Edited, offset 1 and a(1)=2 in programs and b-file by Georg Fischer, Nov 22 2020
Comments