A281409 -id:A281409 - cp's OEIS frontend

A281408 Lexicographically first sequence of distinct terms with the property that each triple of consecutive terms contains a term that divides the difference of the other two terms.

1, 2, 3, 5, 8, 13, 21, 4, 9, 17, 26, 43, 69, 112, 181, 23, 20, 63, 83, 10, 33, 53, 86, 11, 15, 37, 7, 6, 19, 25, 44, 94, 50, 22, 14, 36, 64, 28, 12, 16, 40, 24, 88, 32, 56, 120, 176, 296, 30, 38, 68, 106, 174, 34, 35, 103, 138, 241, 379, 46, 57, 149, 92, 333

Offset: 1

Rémy Sigrist, Jan 21 2017

nonn

			The first terms, alongside the indexes of the terms that divides the difference of the other two terms within the n-th triple of consecutive terms, are:
   n  a(n)    Indexes
  --  ----    -------
   1     1    1, 2
   2     2    1, 2
   3     3    1, 2
   4     5    1, 2
   5     8    1, 2
   6    13    3
   7    21    2
   8     4    1
   9     9    1, 2
  10    17    1, 2
  11    26    1, 2
  12    43    1, 2
  13    69    1, 2
  14   112    3
  15   181    2
  16    23    2
  17    20    1, 2
  18    63    3
  19    83    2
  20    10    1
  21    33    1, 2
  22    53    3
  23    86    3
  24    11    1
  25    15    1

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A281408

Cf. A278962, A281409.

A284148 Lexicographically earliest sequence of nonnegative integers such that a(1)=3 and the sequence is p-periodic mod p for any p > 0.

Original entry on oeis.org

3, 0, 1, 0, 3, 28, 45, 276, 595, 1128, 1953, 3160, 4851, 264540, 190333, 254268, 18915, 3366496, 32385, 125391168, 199588483, 64620, 174673821, 5039370820, 1784859363, 16908230328, 165025, 34420237176, 58409997075, 1367074573228, 2294838551853, 15289788305820

Offset: 1

Rémy Sigrist, Mar 21 2017

nonn

The initial term a(1)=3 seems to be the least one that leads to a sequence that does not have a polynomial closed form.
The first cycles mod p of this sequence are:
p  Cycle of a mod p
-  ----------------
1  0
2  1, 0
3  0, 0, 1
4  3, 0, 1, 0
5  3, 0, 1, 0, 3
6  3, 0, 1, 0, 3, 4
7  3, 0, 1, 0, 3, 0, 3
8  3, 0, 1, 0, 3, 4, 5, 4
9  3, 0, 1, 0, 3, 1, 0, 6, 1
For k>=0, let c_k denote the variant with initial term k.
Naturaly, we have a=c_3.
For some values of k, c_k has a polynomial closed form.
The first such values to be known are:
- k=0: c_0(n) = 0 = A000004(n),
- k=1: c_1(n) = (n-2)^2 = A000290(n-2),
- k=2: c_2(n) = (n-2)*(n-3) = A002378(n-3),
- k=19: c_19(n) = (n-2)*(n^3 - 14*n^2 + 63*n - 88)/2,
- k=20: c_20(n) = (n-2)*(n-3)*(n-5)*(n-6)/2,
- k=22: c_22(n) = (n-2)*(n-3)*(n^2 - 11*n + 32)/2,
- k=40: c_40(n) = (n-2)*(n-3)*(n-5)*(n-6),
- k=172: c_172(n) = (n-2)*(n-3)*(n-5)*(n^3 - 23*n^2 + 172*n - 408)/12.
We notice that c_40 = 2*c_20.
As for A281409, this sequence is the first of a family (of sequences parametrized by their initial term) showing some kind of irregularity.
For k>=0 and n>0, let d_n(k)=c_k(n):
- In particular: d_1(k)=k, and a(n)=d_n(3),
- For any n>1, d_n is periodic.
The cycles for the first d_n (with n>1) are:
n  Cycle of d_n
-  ------------
2  0
3  0, 1
4  0, 4, 2
5  0, 9, 6, 3
6  0, 16, 12, 28, 24, 40, 36, 52, 48, 4
7  0, 25, 20, 45, 40, 5

			By definition, a(1)=3.
a(2) must equal 3 mod 1; a(2)=0 is suitable.
a(3) must equal 3 mod 2 and 0 mod 1; a(3)=1 is suitable.
a(4) must equal 3 mod 3 and 0 mod 2 and 1 mod 1; a(4)=0 is suitable.
a(5) must equal 3 mod 4 and 0 mod 3 and 1 mod 2 and 0 mod 1; a(5)=3 is suitable.

Rémy Sigrist, Table of n, a(n) for n = 1..2000
Rémy Sigrist, PARI program for A284148

Cf. A000004, A000290, A002378, A281409.

cp's OEIS Frontend

A281408 Lexicographically first sequence of distinct terms with the property that each triple of consecutive terms contains a term that divides the difference of the other two terms.

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