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2017-07-01, 14:45   #1
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

953 Posts
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
 fibonacci sequence with coefficient.pdf (70.3 KB, 109 views)

 2017-07-01, 15:46 #2 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3·3,203 Posts 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!
2017-07-02, 16:33   #3
Dr Sardonicus

Feb 2017
Nowhere

5,113 Posts

Quote:
 Originally Posted by MattcAnderson 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.

2017-07-03, 21:04   #4
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

953 Posts

Hi Mersenne forum,

Thank you for the good and constructive replies so far.

See attachment.

Regards,
Matt
Attached Files
 temp.txt (33 Bytes, 105 views)

 2017-07-05, 08:05 #5 carpetpool     "Sam" Nov 2016 23·41 Posts 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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