From: israel@math.ubc.ca (Robert Israel) Subject: Re: Non- and part-symmetric Finonacci-like series Date: 21 Dec 2000 20:52:32 GMT Newsgroups: sci.math Summary: [missing] In article , Axel Harvey wrote: >However, there are sequences which seem to go on for a long time >without any symmetric elements. FWIW, I lack any powerful arithmetic >tool that would allow empirical "verification." Among the sequences >which seem to have this totally asymmetric property, my favourites are >the ones I call > >the Priamic ...9, -5, 4, -1, 3, 2, 5, 7, 12, 19, 31, 50, 81... and > >the Pythagorean ...49, -30, 19, -11, 8, -3, 5, 2, 7, 9, 16, 25, 41... > >Is there a way to show that a sequence will have no symmetric elements? Consider a sequence u_n with u_n=u_{n-1}+u_{n-2}. It is well known that u_n = F_n u_1 + F_{n-1} u_0, where F_n is the n'th Fibonacci number. If u_n = u_0, this means that (1-F_{n-1}) u_0 = F_n u_1 or u_1/u_0 = (1 - F_{n-1})/F_n (assuming n <> 0 and u_0 <> 0). Thus for each integer n > 1 there is, up to scaling (u'_j = c u_j) and translation (u'_j = u_{j+k}), a unique sequence that has two equal elements n places apart. And this sequence is U_j = F_j + (-1)^j F_{n-j} = F_j - (-1)^n F_{j-n}, which has U_0 = U_n = F_n. If n = 4m is divisible by 4, then U_{2m} = F_{2m} + (-1)^{2m} F_{2m} = 2 F_{2m} and U_{2m+1} = F_{2m+1} - F_{2m-1} = F_{2m}, so U_j is F_{2m} times a translate of the Lucas sequence. If n = 4m+2, then U_{2m+1} = 0 and U_j is a multiple of a translate of the Fibonacci sequence. If n=1 or 3, we also have a multiple of a translate of the Fibonacci sequence. If n is odd and greater than 3, we have what you called "unique symmetric elements", with U_j = F_j + F_{j-n}. Thus your example -27, 17, -10, 7, -3, 4, 1, 5, 6, 11, 17, 28 is u_j = (F_j + F_{j-9})/2 (with u_0 at the first 17). In these cases if U_j = U_k we would have F_j + F_{j-n} = F_k + F_{k-n}. I think that if n is odd and greater than 3, this is only possible if j=k or {j,k} = {0,n}. This should be easy to prove. In fact the equation could be put in the form (+/-) F_|j| (+/-) F_|j-n| (+/-) F_|k| (+/-) F_|k-n| = 0 and the only equations (+/-) F_a (+/-) F_b (+/-) F_c (+/-) F_d = 0 where a >= b >= c >= d >= 0 are with a=b, b+1 or b+2. Some rather tedious examination of cases should yield the result. If this works, your "unique symmetric elements" really are unique. Similarly for u_n = - u_0 we have u_1/u_0 = (-1 - F_{n-1})/F_n and the sequence is V_j = - F_j + (-1)^j F_{n-j} = - F_j - (-1)^n F_{j-n}, with V_0 = -V_n = F_n. Again, the Fibonacci and Lucas sequences take the even cases as well as n=1 and 3, leaving the odd n > 3 with "unique antisymmetric elements". As for the question of whether your "Priamic" or "Pythagorean" sequences have symmetric or antisymmetric elements, note that the element of least absolute value in U_j = F_j + F_{j-n} (with n odd) is either U_{(n-1)/2} or U_{(n+1)/2}. If n=4m+1, U_{(n-1)/2} = U_{2m} = F_{2m+2} and U{(n+1)/2} = U_{2m+1} = F_{2m-1}, while if n=4m-1, U_{(n-1)/2} = U_{2m-1} = -F_{2m-2} and U_{(n+1)/2} = U_{2m} = F_{2m+1}. Similarly for the antisymmetric case: if n=4m+1, V_{2m} = F_{2m-1} and V_{2m+1} = -F_{2m+2}, while if n=4m-1, V_{2m-1} = -F_{2m+1} and V_{2m} = -F_{2m-2}. So to tell whether a given sequence has symmetric or antisymmetric elements, we just need to look for the element of least absolute value and its nearest neighbours, and see whether the ratio of their absolute values is that of two Fibonacci numbers 3 positions apart. For the "Priamic", 1/4 = 2/8 = F_3/F_6 but with m=2, n=9, U_j has the wrong signs (8,2 as opposed to 4,-1) while V_j has the wrong order (2,-8 as opposed to 4,-1). On the other hand, 1/3 = F_1/F_4 and with m=1, n=5, V_j is -1 times your "Priamic" sequence, with the antisymmetric elements 5 and -5. For your "Pythagorean", neither 2/5 nor 2/7 is a ratio of Fibonacci numbers 3 positions apart, so there are no symmetric or antisymmetric elements. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2