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A068380 Engel expansion of sinh(1/3).

%I A068380 #42 May 30 2025 23:57:04
%S A068380 3,54,180,378,648,990,1404,1890,2448,3078,3780,4554,5400,6318,7308,
%T A068380 8370,9504,10710,11988,13338,14760,16254,17820,19458,21168,22950,
%U A068380 24804,26730,28728,30798,32940,35154,37440,39798,42228,44730,47304,49950,52668,55458,58320,61254
%N A068380 Engel expansion of sinh(1/3).
%C A068380 Cf. A006784 for the definition of the Engel expansion.
%C A068380 The MathWorld link mentions the closed form of the Engel expansion of sinh(1). - _Georg Fischer_, Nov 22 2020
%H A068380 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>.
%H A068380 Wikipedia, <a href="https://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>.
%H A068380 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A068380 a(n) = 18*(n*(2*n-3)+1) for n > 1, a(1)=3. - _Ralf Stephan_, Sep 03 2003
%F A068380 From _Colin Barker_, Apr 13 2012: (Start)
%F A068380 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
%F A068380 G.f.: 3*x*(x^3-9*x^2-15*x-1)/(x-1)^3. (End)
%F A068380 From _Amiram Eldar_, May 05 2025: (Start)
%F A068380 Sum_{n>=1} 1/a(n) = (4-log(2))/9.
%F A068380 Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 - Pi/36 - log(2)/18. (End)
%F A068380 From _Elmo R. Oliveira_, May 29 2025: (Start)
%F A068380 E.g.f.: 3*(6*exp(x)*(1 - x + 2*x^2) + (x - 6)).
%F A068380 a(n) = 9*A002943(n-1) for n >= 2. (End)
%e A068380 sinh(1/3) = 1/3 + 1/(3*54) + 1/(3*54*180) + 1/(3*54*180*378) + 1/(3*54*180*378*648) + ... = 0.33954055725615013910126061...
%t A068380 LinearRecurrence[{3, -3, 1}, {3, 54, 180, 378}, 50]
%o A068380 (PARI) a(n)=if(n<=1, 3, 18*(n*(2*n-3)+1));
%o A068380 (PARI) my(x='x+O('x^43)); Vec(3*x*(1+15*x+9*x^2-x^3)/(1-x)^3) \\ _Elmo R. Oliveira_, May 29 2025
%Y A068380 Cf. A006784, A002943, A068377, A068379.
%K A068380 easy,nonn
%O A068380 1,1
%A A068380 _Benoit Cloitre_, Mar 03 2002
%E A068380 Edited, offset 1 and a(1)=3 by _Georg Fischer_, Nov 23 2020
%E A068380 More terms from _Elmo R. Oliveira_, May 29 2025