|
| |
|
|
A087798
|
|
a(n) = 9a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 9, a(n) = [(9+sqrt(85))/2]^n + [(9-sqrt(85))/2]^n.
|
|
4
| |
|
|
2, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, 432083484, 3936182123, 35857722591, 326655685442, 2975758891569, 27108485709563, 246952130277636, 2249677658208287, 20494051054152219, 186696137145578258
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,1
|
|
|
COMMENTS
| Harmonious sequence, built on the number 9.10977...
a(n+1)/a(n) converges to (9+sqrt(85))/2 = 9.10977... a(0)/a(1)=2/9; a(1)/a(2)=9/83; a(2)/a(3)=83/756; a(3)/a(4)=756/6887; ... etc. Lim a(n)/a(n+1) as n approaches infinity = 0.1097722... = 2/(9+sqrt(85)) = (sqrt(85)-9)/2.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
For more information about this type of recurrence follow the Khovanova link and see A054413 and A086902.
(End)
|
|
|
LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
|
|
|
FORMULA
| G.f.: (2-9*x)/(1-9*x-x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 02 2008]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
a(2n+1) = 9*A097840(n), a(2n) = A099373(n).
a(3n+1) = A041150(5n), a(3n+2) = A041150(5n+3), a(3n+3) = 2*A041150(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A087798(n) + A099371(n)*sqrt(85))/2.
Limit(A087798(n)/A099371(n), n=infinity) = sqrt(85).
(End)
|
|
|
EXAMPLE
| a(4) = 6887 = 9a(3) + a(2) = 9*756 + 83 = [(9+sqrt(85))/2]^4 + [(9-sqrt(85))/2]^4 = 6886.9998547 + 0.0001452 = 6887.
|
|
|
CROSSREFS
| Cf. A014511.
Sequence in context: A123570 A006040 A067309 * A113146 A069234 A086929
Adjacent sequences: A087795 A087796 A087797 * A087799 A087800 A087801
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Nikolay V. Kosinov, Dmitry V. Poljakov (kosinov(AT)unitron.com.ua), Oct 10 2003
|
|
|
EXTENSIONS
| More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 06 2003
|
| |
|
|
