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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

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OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..38.

Eric Weisstein's World of Mathematics, Engel Expansion.

Wikipedia, Engel Expansion.

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

FORMULA

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

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. - Colin Barker, Apr 13 2012

EXAMPLE

1/3 + 1/(3*54) + 1/(3*54*180) + 1/(3*54*180*378) + 1/(3*54*180*378*648) ...

= 0.3395405572560086...

sinh(1/3) = 0.33954055725615013910126061...

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A006784, A068379.

Sequence in context: A344424 A045481 A275566 * A174782 A345074 A119294

Adjacent sequences: A068377 A068378 A068379 * A068381 A068382 A068383

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, Mar 03 2002

EXTENSIONS

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

STATUS

approved