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

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10001

Eric W. Weisstein, Engel Expansion

Wikipedia, Engel Expansion

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

Cf. A006784, A068377, A068380, A334367.

Sequence in context: A119060 A119050 A119068 * A136280 A123831 A138648

Adjacent sequences: A068376 A068377 A068378 * A068380 A068381 A068382

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