login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A142985 a(1) = 1, a(2) = 6, a(n+2) = 6*a(n+1) + (n+1)*(n+2)*a(n). 5
1, 6, 42, 324, 2784, 26424, 275472, 3132576, 38629440, 513708480, 7331489280, 111798455040, 1814503057920, 31234337164800, 568451665152000, 10906950910464000, 220060558384128000, 4657890328906752000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is the case m = 3 of the general recurrence a(1) = 1, a(2) = 2*m, a(n+2) = 2*m*a(n+1)+(n+1)*(n+2)*a(n), which arises when accelerating the convergence of a certain series for the constant log(2). See A142983 for remarks on the general case.

REFERENCES

Bruce C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

FORMULA

a(n) = n!*p(n+1)*sum {k = 1..n} (-1)^(k+1)/(p(k)*p(k+1)), where p(n) = (2*n^3+n)/3 = A005900(n). Recurrence: a(1) = 1, a(2) = 6, a(n+2) = 6*a(n+1)+(n+1)*(n+2)*a(n). The sequence b(n):= n!*p(n+1) satisfies the same recurrence with b(1) = 6, b(2) = 38. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(6 +1*2/(6 +2*3/(6 +3*4/(6 +...+(n-1)*n/6)))), for n >=2. The behavior of a(n) for large n is given by lim n -> infinity a(n)/b(n) = sum {k = 1..inf} (-1)^(k+1)/(p(k)*p(k+1)) = 1/(6 +1*2/(6 +2*3/(6 +3*4/(6 +...+n*(n+1)/(6 +...))))) = 6*log(2) - 4, where the final equality follows by a result of Ramanujan (see [Berndt, Chapter 12, Entry 32(i)]).

MAPLE

p := n -> (2*n^3+n)/3: a := n -> n!*p(n+1)*sum ((-1)^(k+1)/(p(k)*p(k+1)), k = 1..n): seq(a(n), n = 1..20);

MATHEMATICA

RecurrenceTable[{a[1]==1, a[2]==6, a[n]==6a[n-1]+(n-1)n*a[n-2]}, a, {n, 20}] (* Harvey P. Dale, Sep 20 2013 *)

PROG

(Haskell)

a142985 n = a142985_list !! (n-1)

a142985_list = 1 : 6 : zipWith (+)

                       (map (* 6) $ tail a142985_list)

                       (zipWith (*) (drop 2 a002378_list) a142985_list)

-- Reinhard Zumkeller, Jul 17 2015

CROSSREFS

Cf. A005900, A142983, A142984, A142986, A142987.

Cf. A002378.

Sequence in context: A153293 A145301 A107266 * A118351 A033296 A218755

Adjacent sequences:  A142982 A142983 A142984 * A142986 A142987 A142988

KEYWORD

easy,nonn

AUTHOR

Peter Bala, Jul 17 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 23 09:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)