A068380 Engel expansion of sinh(1/3).

3, 54, 180, 378, 648, 990, 1404, 1890, 2448, 3078, 3780, 4554, 5400, 6318, 7308, 8370, 9504, 10710, 11988, 13338, 14760, 16254, 17820, 19458, 21168, 22950, 24804, 26730, 28728, 30798, 32940, 35154, 37440, 39798, 42228, 44730, 47304, 49950, 52668, 55458, 58320, 61254

Offset: 1

Author

Benoit Cloitre, Mar 03 2002

Keywords

• easy
• nonn

Comments

Cf. A006784 for the definition of the Engel expansion.

The MathWorld link mentions the closed form of the Engel expansion of sinh(1). - Georg Fischer, Nov 22 2020

Examples

			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...
		

Links

• Eric Weisstein's World of Mathematics, Engel Expansion.
• Wikipedia, Engel Expansion.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A006784, A002943, A068377, A068379.

Programs

Mathematica

LinearRecurrence[{3, -3, 1}, {3, 54, 180, 378}, 50]

PARI

a(n)=if(n<=1, 3, 18*(n*(2*n-3)+1));

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

Formula

a(n) = 18*(n*(2*n-3)+1) for n > 1, a(1)=3. - Ralf Stephan, Sep 03 2003

From Colin Barker, Apr 13 2012: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.

G.f.: 3*x*(x^3-9*x^2-15*x-1)/(x-1)^3. (End)

From Amiram Eldar, May 05 2025: (Start)

Sum_{n>=1} 1/a(n) = (4-log(2))/9.

Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 - Pi/36 - log(2)/18. (End)

From Elmo R. Oliveira, May 29 2025: (Start)

E.g.f.: 3*(6*exp(x)*(1 - x + 2*x^2) + (x - 6)).

a(n) = 9*A002943(n-1) for n >= 2. (End)

Extensions

Edited, offset 1 and a(1)=3 by Georg Fischer, Nov 23 2020

More terms from Elmo R. Oliveira, May 29 2025