This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A068379 #57 May 30 2025 23:57:13
%S A068379 2,24,80,168,288,440,624,840,1088,1368,1680,2024,2400,2808,3248,3720,
%T A068379 4224,4760,5328,5928,6560,7224,7920,8648,9408,10200,11024,11880,12768,
%U A068379 13688,14640,15624,16640,17688,18768,19880,21024,22200,23408,24648,25920,27224,28560
%N A068379 Engel expansion of sinh(1/2).
%C A068379 Cf. A006784 for Engel expansion definition.
%C A068379 The MathWorld link mentions the closed form of the Engel expansion of sinh(1) = A068377. - _Georg Fischer_, Nov 22 2020
%H A068379 Vincenzo Librandi, <a href="/A068379/b068379.txt">Table of n, a(n) for n = 1..10001</a>
%H A068379 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>.
%H A068379 Wikipedia, <a href="https://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>.
%H A068379 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A068379 a(n) = 8*(n*(2*n-3)+1) for n > 1, a(1)=2.
%F A068379 From _Colin Barker_, Apr 13 2012: (Start)
%F A068379 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
%F A068379 G.f.: 2*x*(1+9*x+7*x^2-x^3)/(1-x)^3. (End)
%F A068379 From _Amiram Eldar_, May 05 2025: (Start)
%F A068379 Sum_{n>=1} 1/a(n) = (3-log(2))/4.
%F A068379 Sum_{n>=1} (-1)^(n+1)/a(n) = 3/4 - Pi/16 - log(2)/8. (End)
%F A068379 From _Elmo R. Oliveira_, May 29 2025: (Start)
%F A068379 E.g.f.: 2*(4*exp(x)*(1 - x + 2*x^2) + (x - 4)).
%F A068379 a(n) = 2*A033586(n-1) for n >= 2.
%F A068379 a(n) = 4*A002943(n-1) for n >= 2. (End)
%e A068379 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.
%t A068379 Table[If[n==1, 2, 8*(n*(2*n-3)+1)], {n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 30 2012 *)
%t A068379 LinearRecurrence[{3,-3,1},{2,24,80,168},50] (* _Harvey P. Dale_, Mar 21 2017 *)
%o A068379 (Magma) [2] cat [8*(n*(2*n-3)+1): n in [2..50]]; // _Vincenzo Librandi_, Jan 31 2012
%o A068379 (PARI) a(n)=if(n<=1,2,8*(n*(2*n-3)+1)) \\ _Charles R Greathouse IV_, Jan 31 2012
%Y A068379 Cf. A002943, A006784, A033586, A068377, A068380, A334367.
%K A068379 easy,nonn
%O A068379 1,1
%A A068379 _Benoit Cloitre_, Mar 03 2002
%E A068379 Edited, offset 1 and a(1)=2 in programs and b-file by _Georg Fischer_, Nov 22 2020