20170701, 14:45  #1 
"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 
20170701, 15:46  #2 
"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! 
20170702, 16:33  #3  
Feb 2017
Nowhere
5,113 Posts 
Quote:
For a monic quadratic x^2  a*x  b, the Lucas and Fibonaccilike sequences are L_{0} = 2, L_{1} = a, L_{k+2} = L_{k+1} + b*L_{k} F_{0} = 0, F_{1} = 1, F_{k+2} = F_{k+1} + b*F_{k} 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 Qlinear combination of L_{n} and F_{n}. With a = 1 and b = 10, we have L_{0} = 2, L_{1} = 1, L_{2} = 21, L_{3} = 31, L_{4} = 251... F_{0} = 0, F_{1} = 1, F_{2} = 1, F_{3} = 11, F_{4} = 21, ... With a = 1, b = 10, the example sequence may be written ld(n) = (31*F_{n}  L_{n})/5; ld(1) = 6, ld(2) = 2, ld(3) = 62, etc The coefficients may be found easily by "reverseengineering" the value ld(0) = 2/5, then noting that F_{0} = 0 and L_{0} = 2, making the coefficient of L_{n} 1/5. The coefficient of L_{n} 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. 

20170703, 21:04  #4 
"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 
20170705, 08:05  #5 
"Sam"
Nov 2016
2^{3}·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(n1)*a + F(n2) and Lucas Like sequences: L(1) = 1 L(2) = a^2+2 L(3) = a^2+3 L(n) = L(n1) + L(n2) if n is odd. L(n) = (a^2+1)*L(n1)  L(n3) if n is even. 
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