```Date: Apr 23, 2013 9:53 AM
Author: Morten Leikvoll
Subject: circular sequence problem

I'm posting this again to a different group. Im not really a big matematician but need a solution for this interesting problem.Given b,c both constant positive integers,using computer style functions:(x mod y)  - modulo/modulus (remainder of x divided by y) function(x div y)    - integer of x divided by y functionI got this recursive (circular) sequence a() where n runs from 0 to b*c-1. (The sequence a() repeats so that a(n+b*c,m)=a(n,m))for m=0 : a(n,m)=nfor m>0 : a(n,m)=( a(n,m-1) mod b )*c + ( a(n,m-1) div b) mod cDoes there exist a general formula for:a(n,m) = f(..) taking any of the parameters n, a(n-1,m), m, b, c, but not a(n,m-1)?As a second challenge for curiosity. There exist an integer R such that a(n,m+R)=a(n,m). What is the R?Other characteristics:-R is symmetric around b,c axis (swapping b,c values give same R)-For all b=c => R=2-From experimentation, it appears that for any n in range 2 to b*c-2, only R'=R gives a(n,m+R')=a(n,m).Thanks for any input. Below I add some sample numbers from the sequence.------------------------Example series for b=4,c=3, showing sequence for n=0..11 and m=0..5a(n,0)=0,1,2,3,4,5,6,7,8,9,10,11a(n,1)=0,4,8,1,5,9,2,6,10,3,7,11a(n,2)=0,5,10,4,9,3,8,2,7,1,6,11a(n,3)=0,9,7,5,3,1,10,8,6,4,2,11a(n,4)=0,3,6,9,1,4,7,10,2,5,8,11a(n,5)=0,1,2,3,4,5,6,7,8,9,10,11This sequence repeats for every 5th value of m (R=5)Example series for b=10,c=3, showing sequence for n=0..29 and m=0..28a(n,0)=0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29a(n,1)=0,10,20,1,11,21,2,12,22,3,13,23,4,14,24,5,15,25,6,16,26,7,17,27,8,18,28,9,19,29a(n,2)=0,13,26,10,23,7,20,4,17,1,14,27,11,24,8,21,5,18,2,15,28,12,25,9,22,6,19,3,16,29a(n,3)=0,14,28,13,27,12,26,11,25,10,24,9,23,8,22,7,21,6,20,5,19,4,18,3,17,2,16,1,15,29a(n,4)=0,24,19,14,9,4,28,23,18,13,8,3,27,22,17,12,7,2,26,21,16,11,6,1,25,20,15,10,5,29a(n,5)=0,8,16,24,3,11,19,27,6,14,22,1,9,17,25,4,12,20,28,7,15,23,2,10,18,26,5,13,21,29a(n,6)=0,22,15,8,1,23,16,9,2,24,17,10,3,25,18,11,4,26,19,12,5,27,20,13,6,28,21,14,7,29a(n,7)=0,17,5,22,10,27,15,3,20,8,25,13,1,18,6,23,11,28,16,4,21,9,26,14,2,19,7,24,12,29a(n,8)=0,25,21,17,13,9,5,1,26,22,18,14,10,6,2,27,23,19,15,11,7,3,28,24,20,16,12,8,4,29a(n,9)=0,18,7,25,14,3,21,10,28,17,6,24,13,2,20,9,27,16,5,23,12,1,19,8,26,15,4,22,11,29a(n,10)=0,6,12,18,24,1,7,13,19,25,2,8,14,20,26,3,9,15,21,27,4,10,16,22,28,5,11,17,23,29a(n,11)=0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,1,3,5,7,9,11,13,15,17,19,21,23,25,27,29a(n,12)=0,20,11,2,22,13,4,24,15,6,26,17,8,28,19,10,1,21,12,3,23,14,5,25,16,7,27,18,9,29a(n,13)=0,26,23,20,17,14,11,8,5,2,28,25,22,19,16,13,10,7,4,1,27,24,21,18,15,12,9,6,3,29a(n,14)=0,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,29a(n,15)=0,19,9,28,18,8,27,17,7,26,16,6,25,15,5,24,14,4,23,13,3,22,12,2,21,11,1,20,10,29a(n,16)=0,16,3,19,6,22,9,25,12,28,15,2,18,5,21,8,24,11,27,14,1,17,4,20,7,23,10,26,13,29a(n,17)=0,15,1,16,2,17,3,18,4,19,5,20,6,21,7,22,8,23,9,24,10,25,11,26,12,27,13,28,14,29a(n,18)=0,5,10,15,20,25,1,6,11,16,21,26,2,7,12,17,22,27,3,8,13,18,23,28,4,9,14,19,24,29a(n,19)=0,21,13,5,26,18,10,2,23,15,7,28,20,12,4,25,17,9,1,22,14,6,27,19,11,3,24,16,8,29a(n,20)=0,7,14,21,28,6,13,20,27,5,12,19,26,4,11,18,25,3,10,17,24,2,9,16,23,1,8,15,22,29a(n,21)=0,12,24,7,19,2,14,26,9,21,4,16,28,11,23,6,18,1,13,25,8,20,3,15,27,10,22,5,17,29a(n,22)=0,4,8,12,16,20,24,28,3,7,11,15,19,23,27,2,6,10,14,18,22,26,1,5,9,13,17,21,25,29a(n,23)=0,11,22,4,15,26,8,19,1,12,23,5,16,27,9,20,2,13,24,6,17,28,10,21,3,14,25,7,18,29a(n,24)=0,23,17,11,5,28,22,16,10,4,27,21,15,9,3,26,20,14,8,2,25,19,13,7,1,24,18,12,6,29a(n,25)=0,27,25,23,21,19,17,15,13,11,9,7,5,3,1,28,26,24,22,20,18,16,14,12,10,8,6,4,2,29a(n,26)=0,9,18,27,7,16,25,5,14,23,3,12,21,1,10,19,28,8,17,26,6,15,24,4,13,22,2,11,20,29a(n,27)=0,3,6,9,12,15,18,21,24,27,1,4,7,10,13,16,19,22,25,28,2,5,8,11,14,17,20,23,26,29a(n,28)=0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29This sequence repeats for every 28th value of m (R=28)Also, note that b=c implies R=2Some R's from an array of b,cb\c  1   2   3   4   5   6   7   8---|-------------------------------1  | 1   1   1   1   1   1   1   12  | 1   2   4   3   6   10  12  43  | 1   4   2   5   6   16  4   114  | 1   3   5   2   9   11  9   55  | 1   6   6   9   2   14  16  46  | 1   10  16  11  14  2   40  237  | 1   12  4   9   16  40  2   208  | 1   4   11  5   4   23  20  29  | 1   8   3   6   5   26  15  3510 | 1   18  28  6   42  58  22  1311 | 1   6   8   7   18  12  6   2812 | 1   11  12  23  29  35  41  1213 | 1   20  18  4   16  10  12  1714 | 1   18  8   10  22  82  96  1215 | 1   28  10  29  36  88  12  816 | 1   5   23  3   39  9   9   717 | 1   10  20  33  6   10  29  1218 | 1   12  52  35  44  106 20  2019 | 1   36  6   10  46  112 10  520 | 1   12  29  39  30  16  69  5221 | 1   20  30  41  4   25  24  8322 | 1   14  12  14  27  130 48  2023 | 1   12  16  6   18  136 4   2024 | 1   23  35  18  48  60  83  9525 | 1   21  18  15  3   37  7   3326 | 1   8   30  51  42  6   12  2227 | 1   52  4   53  22  22  46  2828 | 1   20  41  18  69  83  12  3729 | 1   18  42  22  12  43  100 1030 | 1   58  88  12  37  178 30  11931 | 1   60  22  10  30  4   18  1232 | 1   6   36  7   52  19  37  833 | 1   12  42  65  20  14  44  131Some R's from bigger b,cb=840,c=2409 => R=1010352b=841,c=2409 => R=4522b=2363,c=2414 => R=47580b=2364,c=2414 => R=846
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