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Engel expansion of sinh(1/2).
2

%I #48 Sep 08 2022 08:45:05

%S 2,24,80,168,288,440,624,840,1088,1368,1680,2024,2400,2808,3248,3720,

%T 4224,4760,5328,5928,6560,7224,7920,8648,9408,10200,11024,11880,12768,

%U 13688,14640,15624,16640,17688,18768,19880,21024,22200,23408,24648,25920,27224

%N Engel expansion of sinh(1/2).

%C Cf. A006784 for Engel expansion definition.

%C The MathWorld link mentions the closed form of the Engel expansion of sinh(1) = A068377. - _Georg Fischer_, Nov 22 2020

%H Vincenzo Librandi, <a href="/A068379/b068379.txt">Table of n, a(n) for n = 1..10001</a>

%H Eric W. Weisstein, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(1) = 2, a(n) = 8*(n*(2*n-3)+1) for n>1.

%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3. G.f.: x*(2+18*x+14*x^2-2*x^3)/(1-x)^3. - _Colin Barker_, Apr 13 2012

%e 1/2 + 1/(2*24) + 1/(2*24*80) + 1/(2*24*80*168) + 1/(2*24*80*168*288) ...

%e = 0.5210953054814953...

%e sinh(1/2) = 0.52109530549374736162242562641... = A334367.

%t Table[If[n==1, 2, 8*(n*(2*n-3)+1)], {n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 30 2012 *)

%t LinearRecurrence[{3,-3,1},{2,24,80,168},50] (* _Harvey P. Dale_, Mar 21 2017 *)

%o (Magma) [2] cat [8*(n*(2*n-3)+1): n in [2..50]]; // _Vincenzo Librandi_, Jan 31 2012

%o (PARI) a(n)=if(n<=1,2,8*(n*(2*n-3)+1)) \\ _Charles R Greathouse IV_, Jan 31 2012

%Y Cf. A006784, A068377, A068380, A334367.

%K easy,nonn

%O 1,1

%A _Benoit Cloitre_, Mar 03 2002

%E Edited, offset 1 and a(1)=2 in programs and b-file by _Georg Fischer_, Nov 22 2020