This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A208125 #68 Dec 15 2019 17:01:02
%S A208125 1,1,3,10,41,46,277,566,1601,1834,18341,17802,213625,230870,225365,
%T A208125 465790,7452641,8129874,146337733
%N A208125 Number of distinct n-th rows in arrays whose columns are running modulus recurrence sequences.
%C A208125 For each positive integer b, construct an array of integers with columns indexed by {0,1,2,3,...} and rows indexed by {1,2,3,4,...}. Identify the entry of the n-th row and k-th column by z_{n,k}(b). Set z_{1,k}(b) = 0 for all k. Compute z_{n,k}(b) = (b*z_{n-1,k}(b) + k) mod n, always using least nonnegative residues. The columns of these arrays are RuMoR (Running Modulus Recurrence) sequences. The sequence given is the number of distinct n-th rows among all arrays.
%C A208125 In the Mathematica dynamic program that accompanies the paper by Dearden et al. (2013), the output table is the transpose of the array alluded in the name of the sequence, the comments above, and the example below. The same holds for the table on pp. 3-4 in the paper (where b = 2 for the table on p. 3 and b = 6 for the table on p. 4). - _Petros Hadjicostas_, Dec 13 2019
%H A208125 B. Dearden, J. Iiams, and J. Metzger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Dearden/dearden3r.html">A Function Related to the Rumor Sequence Conjecture</a>, J. Integer Seq. 14(1) (2011), Article 11.2.3.
%H A208125 B. Dearden, J. Iiams, and J. Metzger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Dearden/dearden4.html">Rumor Arrays</a>, J. Integer Seq. 16 (2013), Article 13.9.3.
%H A208125 B. Dearden and J. Metzger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Dearden/dearden3.html">Running Modulus Recursions</a>, J. Integer Seq. 13(1) (2010), Article 10.1.6.
%e A208125 a(3) = 3. Every distinct row is periodic with a period dividing 6. b=1 generates 0, 2, 2, 1, 1, 0 repeating, b=2 generates 0, 0, 2, 2, 1, 1 repeating, and b=3 generates 0, 1, 2 repeating. All other values of b give one of these.
%e A208125 From _Petros Hadjicostas_, Dec 13 2019: (Start)
%e A208125 Using the dynamic Mathematica program provided with the paper by Dearden et al. (2013) (but taking the transpose of the output table), we see that for all b >= 1 the first row is always 0, 0, 0, 0, ..., so a(1) = 1.
%e A208125 By looking at the second rows, we see that for all b >= 1 the 2nd row is always 0, 1, 0, 1, ..., so a(2) = 1.
%e A208125 By looking at the 3rd rows, we see that for all b with 1 = b mod 3, we get 0, 2, 2, 1, 1, 0 repeating (with period 6); for all b with 2 = b mod 3, we get 0, 0, 2, 2, 1, 1 repeating (with period 6); and with 0 = b mod 3, we get 0, 1, 2 repeating (with period 3). (See also the example above.) Thus, a(3) = 3.
%e A208125 By looking at the 4th rows, we see that for all b with 1 = b mod 12, we get 0, 3, 0, 0, 1, 1, 2, 1, 2, 2, 3, 3 repeating (with period 12); for 2 = b mod 12, we get 0, 1, 2, 3, 2, 3, 2, 3, 0, 1, 0, 1 repeating (with period 12); for 3 = b mod 12, we get 0, 0, 0, 3, 3, 3, 2, 2, 2, 1, 1, 1 repeating (with period 12); for (b mod 12) = 0, 4, or 8, we get 0, 1, 2, 3 repeating (with period 4); for 5 = b mod 12, we get 0, 1, 0, 1, 1, 2, 2, 3, 2, 3, 3, 0 repeating (with period 12); for 6 = b mod 12, we get 0, 3, 2, 3, 2, 1, 2, 1, 0, 1, 0, 3 repeating (with period 12); for 7 = b mod 12, we get 0, 3, 0, 2, 3, 1, 2, 1, 2, 0, 1, 3 repeating (with period 12); for 9 = b mod 12, we get 0, 2, 0, 3, 1, 3, 2, 0, 2, 1, 3, 1 repeating (with period 12); with 10 = b mod 12, we get 0, 1, 2, 1, 2, 1, 2, 3, 0, 3, 0, 3 repeating (with period 12); and for 11 = b mod 12, we get 0, 1, 0, 3, 0, 2, 3, 2, 3, 1, 2 repeating (with period 12). Thus, a(4) = 10. (End)
%t A208125 (* the following code computes a(n) for parameter n - it is slow for n >= 8 *)
%t A208125 x[_, _, 0] = 0; x[b_, c_, n_] := x[b, c, n] = Mod[b*x[b, c, n - 1] + c, n]; Table[mtab = Table[Table[x[b, c, n], {c, 0, Apply[LCM, Range[n]]}], {b, 1, Apply[LCM, Range[n]]}]; Length[Union[mtab]], {n, 6}]
%Y A208125 Cf. A177356, A330411.
%K A208125 nonn,more
%O A208125 1,3
%A A208125 _Joel Iiams_, Mar 28 2012