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A068379
Engel expansion of sinh(1/2).
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
OFFSET
1,1
COMMENTS
Cf. A006784 for Engel expansion definition.
The MathWorld link mentions the closed form of the Engel expansion of sinh(1) = A068377. - Georg Fischer, Nov 22 2020
FORMULA
a(1) = 2, a(n) = 8*(n*(2*n-3)+1) for n>1.
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
EXAMPLE
1/2 + 1/(2*24) + 1/(2*24*80) + 1/(2*24*80*168) + 1/(2*24*80*168*288) ...
= 0.5210953054814953...
sinh(1/2) = 0.52109530549374736162242562641... = A334367.
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 *)
PROG
(Magma) [2] cat [8*(n*(2*n-3)+1): n in [2..50]]; // Vincenzo Librandi, Jan 31 2012
(PARI) a(n)=if(n<=1, 2, 8*(n*(2*n-3)+1)) \\ Charles R Greathouse IV, Jan 31 2012
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, Mar 03 2002
EXTENSIONS
Edited, offset 1 and a(1)=2 in programs and b-file by Georg Fischer, Nov 22 2020
STATUS
approved