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2017-07-01, 14:45   #1
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

22×3×53 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, 53 views)

 2017-07-01, 15:46 #2 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 925910 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

23×181 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

22×3×53 Posts

Hi Mersenne forum,

Thank you for the good and constructive replies so far.

See attachment.

Regards,
Matt
Attached Files
 temp.txt (33 Bytes, 53 views)

 2017-07-05, 08:05 #5 carpetpool     "Sam" Nov 2016 2×163 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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