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Old 2017-07-01, 14:45   #1
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

953 Posts
Default another defined sequence

HI Mersenne forum,

I want to shine a light on the hailstone problem.
Can somebody do a google search for me?
According to Numberphile on YouTube, there is a text book on this subject.
The Wikipedia article is my next step.

Regards,
Matt
Attached Files
File Type: pdf fibonacci sequence with coefficient.pdf (70.3 KB, 109 views)
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Old 2017-07-01, 15:46   #2
Batalov
 
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"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

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You are describing the generalized Lucas sequences.
http://primes.utm.edu/top20/page.php?id=23 (and see other places)

They share many properties with Fib and Lucas (they are partial cases). Wagstaff numbers are a partial case, too!
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Old 2017-07-02, 16:33   #3
Dr Sardonicus
 
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Feb 2017
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Quote:
Originally Posted by MattcAnderson View Post
HI Mersenne forum,

I want to shine a light on the hailstone problem.
Can somebody do a google search for me?
According to Numberphile on YouTube, there is a text book on this subject.
The Wikipedia article is my next step.

Regards,
Matt
(referring to OP's attached file)

For a monic quadratic x^2 - a*x - b, the Lucas- and Fibonacci-like sequences are

L0 = 2, L1 = a, Lk+2 = Lk+1 + b*Lk

F0 = 0, F1 = 1, Fk+2 = Fk+1 + b*Fk

These sequences have divisibility properties similar to those of the original Lucas and Fibonacci numbers.

Any sequence of rational numbers with the same recursion is a Q-linear combination of Ln and Fn.

With a = 1 and b = 10, we have

L0 = 2, L1 = 1, L2 = 21, L3 = 31, L4 = 251...

F0 = 0, F1 = 1, F2 = 1, F3 = 11, F4 = 21, ...

With a = 1, b = 10, the example sequence may be written ld(n) = (31*Fn - Ln)/5; ld(1) = 6, ld(2) = 2, ld(3) = 62, etc

The coefficients may be found easily by "reverse-engineering" the value ld(0) = -2/5, then noting that

F0 = 0 and L0 = 2, making the coefficient of Ln -1/5.

The coefficient of Ln is then easily seen to be 31/5.

I don't know offhand of any particularly "nice" divisibility properties for ld(n).

I also don't know any particular connection with the "hailstone problem," AKA the Collatz conjecture.
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Old 2017-07-03, 21:04   #4
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

953 Posts
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Hi Mersenne forum,

Thank you for the good and constructive replies so far.

See attachment.

Regards,
Matt
Attached Files
File Type: txt temp.txt (33 Bytes, 105 views)
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Old 2017-07-05, 08:05   #5
carpetpool
 
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"Sam"
Nov 2016

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I don't know weather you are describing these sequences:

for an integer a,

Fibonacci Like sequences:

F(1) = 1
F(2) = a
F(n) = F(n-1)*a + F(n-2)

and

Lucas Like sequences:

L(1) = 1
L(2) = a^2+2
L(3) = a^2+3

L(n) = L(n-1) + L(n-2) if n is odd.

L(n) = (a^2+1)*L(n-1) - L(n-3) if n is even.
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