| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
Proposed by Cristian Turcu, London, UK.
For a fixed number A, define a sequence {X[n]: n>= 0} by
3X[n]-sqrt(5X[n]^2+4A^2)
X[0] = 0 and X[n+1] = ------------------------
2
(a) For which A is the sequence X[n] convergent?
(b) For which A are all X[n] integers?
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1669.1 | trying it out | STAR::ABBASI | the poet in me want to rise | Fri Oct 02 1992 10:59 | 16 |
carrying this to few terms gives
0, -A, -3A, -8A, -21A, -55A, -144A,..
anyone sees a pattern other than the recursive one given in .0 ?
>(a) For which A is the sequence X[n] convergent?
from above, i dont see any fixed A making this convergent.
>(b) For which A are all X[n] integers?
any integer A will do .
/Nasser
warning, iam doing this before my morning coffee, so if i just made a
big foolish mistake, this is my excuse.
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| 1669.2 | I agree A seems to have nothing to do with it | DESIR::BUCHANAN | The was not found. | Fri Oct 02 1992 12:41 | 12 |
> carrying this to few terms gives > > 0, -A, -3A, -8A, -21A, -55A, -144A,.. > > anyone sees a pattern other than the recursive one given in .0 ? Looks like the well-known Ioac sequence to me. What? You haven't *heard* of the Ioac sequence? It's just the complement of the equally famous Fbnci sequence. :-) Andrew. | |||||
| 1669.3 | linear recurrence | IOSG::CARLIN | Dick Carlin IOSG, Reading, England | Fri Oct 02 1992 13:15 | 22 |
Rewriting:
x[n+1]^2 - 3x[n+1]x[n] = 4a^2 - 4x[n]^2
similarly:
x[n+2]^2 - 3x[n+2]x[n+1] = 4a^2 - 4x[n+1]^2
eliminate a:
x[n+2](x[n+2] - 3x[n+1]) = x[n](x[n] - 3x[n+1])
add x[n+2]x[n] to each side to force a common factor:
x[n+2](x[n+2] - 3x[n+1] + x[n]) = x[n](x[n+2] - 3x[n+1] + x[n])
so (skipping a little rigour):
x[n+2] - 3x[n+1] + x[n] = 0 and a is somewhat of a red herring.
It needs to be an integer to make x[1] an integer and this relation
guarantees the rest will be. And no convergence. Never.
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