Binets formula. http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html
A single statement using a derived table - but it's limited to the first 72 digits before rounding errors cause errors.
drop table a
create table a(a int, b int)
insert a values (1, 1)
insert a values (2, 1)
-- repeat this next statement as many times as you want...
insert a select max(a)+1, sum(b) from a where a>(select max(a) from a)-2
-- get your results with this statement
select * from a
A few syntax changes might be needed for oracle ("insert into a" instead of "insert a" - I do not have Oracle on my machine so I cannot test it right now)
I guess it would be fastest to generate a fixed lookup table from an external function:
I brushed up some VB(A) code in the archive. It don't use recursion as often seen (which is _extremely_ slow for anything more than a handful of elements); instead it fills a simple array (very fast):
Public Function FibonacciElementDec( _
ByVal lngElements As Long) _
As Variant
' Returns the value of element lngElements in the
' Fibonacci sequence of numbers.
' Max. number returned:
' 50095301248058391139327916261
'
' 2004-10-03. Gustav Brock, Cactus Data, CPH.
Const clngIndexMin As Long = 0
Const clngIndexMax As Long = 139
Dim decValue As Variant
If lngElements < clngIndexMin Or lngElements > clngIndexMax Then
' Not a valid input. Return error.
decValue = CDec(-1)
Else
' Return last number in sequence in array.
decValue = FibonacciSequenceDec(lngElements)(lngElements)
End If
FibonacciElementDec = decValue
End Function
Public Function FibonacciSequenceDec( _
ByVal lngElements As Long) _
As Variant()
' Build and return array with Fibonacci sequence of numbers.
' Count of elements is determined by lngElements.
' Max. number returned in array:
' 50095301248058391139327916261
'
' 2004-10-03. Gustav Brock, Cactus Data, CPH.
' Min. index of sequence per definition.
Const clngIndexMin As Long = 0
' Max. possible index of sequence for datatype Decimal.
Const clngIndexMax As Long = 139
Dim adecSeq() As Variant
Dim lngIndex As Long
If lngElements < clngIndexMin Or lngElements > clngIndexMax Then
' Not a valid input.
Else
' Build and fill array with the Fibonacci sequence of numbers.
ReDim adecSeq(clngIndexMin To lngElements)
For lngIndex = clngIndexMin To lngElements
If lngIndex < 2 Then
' Values of the first two elements are 0 and 1 per definition.
adecSeq(lngIndex) = CDec(lngIndex)
Else
' Value is the sum of the two preceding numbers.
adecSeq(lngIndex) = CDec(adecSeq(lngIndex - 2)) + CDec(adecSeq(lngIndex - 1))
End If
'' Debug.Print adecSeq(lngIndex);
Next
End If
FibonacciSequenceDec = adecSeq()
End Function
You can create a simple loop - using either the array from the last function or just repetitive calls of the first function, speed is so fast that you hardly can tell the difference - to insert the index and the value in two fields of the table.
Note that the first element (index zero) is zero.
If you need more than 139 elements I have the equivalent functions for datatype Double but note that you will loose absolute accuracy for element index 76 and beyond.
/gustav
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I reread the question - to be accurate - as far as I know, unless you use a SQL Programming language like TSQL or PL/SQL, it is not possible to create a query that recurses and generates rows without some underlying data. As a result, it is technically impossible to generate the sequence with a SINGLE SQL (pure) statement without using some kind of driver table (a table that has enough rows to drive the creation of new records). Of couse any table with a couple of hundred or so rows will do for this - including data dictionary tables like sysobjects. Once you get into the programming languages, you could program it as easily as you would with VB, just using variables (why bother with a table if you are going to generate a few hundred rows based on a simple algorithm?)
A single statement using a derived table - but it's limited to the first 72 digits before rounding errors cause errors.
select cast (
floor((power((cast(1 as decimal(30,0))+sqrt(cast(5 as decimal(30,0)))),id)-
power((cast(1 as decimal(30,0))-sqrt(cast(5 as decimal(30,0)))),id))/
(power(cast(2 as decimal(30,0)),id)*sqrt(cast(5 as decimal(30,0)))))
as decimal(30,0)) fn
FROM
(
select top 72 cast((a0.id + a1.id) as decimal(30,0)) id FROM
(select 0 id union select 1 union select 2 union select 3 union select 4 union
select 5 union select 6 union select 7 union select 8 union select 9) a0,
(select 0 id union select 10 union select 20 union select 30 union select 40 union
select 50 union select 60 union select 70 union select 80 union select 90) a1
) Numbers
Wow, that's very cool. I don't mind using derived tables. I'm already using one. Actually, I just used the term fibonacci numbers because everyone knows what it is. My aplication doesn't actually use fibonacci numbers, but the idea is almost identical. Basically I have a stack and I want add the numbers up so that it follows a similar fibonacci recurrance relation. So the ith index in the stack equals the sum of the digits from zeroth to (i-1)th indices.
In that case it's probably better just to create a permanent table of integers to join against for you nth index lookups
Select top 1000000 id = identity(int,1,1) Into Numbers from sysobjects s1, sysobjects s1, sysobjects s1, sysobjects s1
In general, an actual table as opposed to a derived table will perform better once you get above a few thousand rows because statistics are kept for a real table and derived tables will be converted to unindexed work tables at some point.
What exactly is your algorithm? We may be able to optimize it.
This goes beyond the scope of the question. If you'd like to email me, drop me a line at rcdetert at nospam ucdavis edu and I'll tell you exactly what I'm doing.
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