A177356 -id:A177356 - cp's OEIS frontend

A208125 Number of distinct n-th rows in arrays whose columns are running modulus recurrence sequences.

1, 1, 3, 10, 41, 46, 277, 566, 1601, 1834, 18341, 17802, 213625, 230870, 225365, 465790, 7452641, 8129874, 146337733

Offset: 1

Joel Iiams, Mar 28 2012

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.

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

			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.
From _Petros Hadjicostas_, Dec 13 2019: (Start)
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.
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.
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.
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)

B. Dearden, J. Iiams, and J. Metzger, A Function Related to the Rumor Sequence Conjecture, J. Integer Seq. 14(1) (2011), Article 11.2.3.
B. Dearden, J. Iiams, and J. Metzger, Rumor Arrays, J. Integer Seq. 16 (2013), Article 13.9.3.
B. Dearden and J. Metzger, Running Modulus Recursions, J. Integer Seq. 13(1) (2010), Article 10.1.6.

Cf. A177356, A330411.

(* the following code computes a(n) for parameter n - it is slow for n >= 8 *)
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}]

A330411 a(n) is the index of the first 0 term in the rumor sequence with initial 0th term 1 and parameters b = 3 and n.

Original entry on oeis.org

2, 1, 3, 5, 4, 24, 2, 22, 21, 20, 19, 6, 17, 16, 15, 14, 4, 6, 20, 22, 99, 8, 97, 3, 95, 94, 93, 92, 13, 393, 89, 391, 9, 389, 85, 84, 83, 82, 384, 80, 382, 60, 59, 58, 75, 377, 376, 375, 53, 373, 372, 68, 370, 66, 368, 796, 24, 365, 793, 363, 362, 361, 789, 788, 787, 786, 14, 784

Offset: 1

Petros Hadjicostas, Dec 13 2019

nonn

"A rumor sequence (running modulus recurrence sequence) is defined as follows: fix integer parameters b > 1 and n > 0. Set z[0] = any integer, and, for k > 0, define z[k] to be the least nonnegative residue of b*z[k-1] modulo (k+n). The rumor sequence conjecture states that all such rumor sequences are eventually 0." [Copied from the comments for A177356]

			For n = 1, we have z[0] = 1, z[1] = 1, and z[2] = 0, so a(1) = 2.
For n = 2, we have z[0] = 1 and z[1] = 0, so a(2) = 1.
For n = 3, we have z[0] = 1, z[1] = 3, z[2] = 4, and z[3] = 0, so a(3) = 3.
For n = 4, we have z[0] = 1, z[1] = 3, z[2] = 3, z[3] = 2, z[4] = 6, and z[5] = 0, so a(4) = 5.
For n = 5, we have z[0] = 1, z[1] = 3, z[2] = 2, z[3] = 6, and z[4] = 0, so a(5) = 4.

B. Dearden, J. Iiams, and J. Metzger, Rumor Arrays, J. Integer Seq. 16 (2013), #13.9.3.
B. Dearden and J. Metzger, Running Modulus Recursions, J. Integer Seq. 13(1) (2010), #10.1.6.

Cf. A177356, A208125.

For[n=1, n<50, n++, k=0; Clear[z]; z[0]=1; z[k_]:=z[k]=Mod[3z[k-1], k+n];
While[z[k]>0, k++]; Print[k]; ] (* Petros Hadjicostas, Dec 14 2019, modification of a program in A177356 *)

a(n) = inf{m > 0 | z[0] = 1, z[m] = 0, and z[k] = (3*z[k-1] mod (k + n)) for k = 1..m}.

A348667 a(n) is the index of the first 0 term in the following sequence: X(1)=1, X(i+1) = (X(i)+i) mod (n+i). a(n)=-1 if there are no zeros in the X sequence.

Original entry on oeis.org

2, 2, 3, 19, 4, 16, 7, 5, 84, 467, 21, 6, 51, 134, 37, 75, 7, 81, 113, 16, 74, 403, 8, 52, 12, 125, 25, 163, 318, 9, 305, 31, 53, 17, 169, 14, 60, 10, 66, 36, 3500, 15, 22, 19, 42, 38, 11, 34, 113, 48, 130, 208, 994, 1033, 468, 17, 12, 53, 307, 623, 45, 173, 48

Offset: 0

Alex Ratushnyak, Oct 28 2021

nonn

Conjecture: a(n) > 0.

Note that some terms are unexpectedly large, for example a(1059) = 62573802.

Cf. A177356.

a[n_] := Module[{x, k = 1}, x[1] = 1; x[j_] := x[j] = Mod[x[j - 1] + j - 1, n + j - 1]; While[x[k] != 0, k++]; k]; Array[a, 63, 0] (* Amiram Eldar, Oct 29 2021 *)

a(n) = my(x=1, i=1); while(x, x = (x+i) % (n+i); i++); i; \\ Michel Marcus, Oct 29 2021

for n in range(1060):
  i = x = 1
  while x:
    x = (x+i) % (n+i)
    i += 1
  print(i, end=', ')

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A208125 Number of distinct n-th rows in arrays whose columns are running modulus recurrence sequences.

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A330411 a(n) is the index of the first 0 term in the rumor sequence with initial 0th term 1 and parameters b = 3 and n.

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A348667 a(n) is the index of the first 0 term in the following sequence: X(1)=1, X(i+1) = (X(i)+i) mod (n+i). a(n)=-1 if there are no zeros in the X sequence.

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