A336684 Irregular triangle in which row n lists residues k found in the sequence Lucas(i) mod n.

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 6, 7, 8, 9, 0, 1, 2, 3, 4, 7, 10, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5

Offset

1,6

Comments

For row n, it is sufficient to take the union of A000032(i) mod n for 0 <= i <= A106291(n - 1), since the Lucas numbers are cyclical mod n.

Row n contains the Lucas number k < n, and k such that (n + k) is a Lucas number.

Row n for n in A224482 is complete, i.e., it contains all residues k (mod n). This includes n that is a perfect power of 3.

Links

Table of n, a(n) for n=1..85.

Formula

A066981(n) = length of row n.

A223487(n) = n - A066981(n) = number of residues missing from row n.

A224482(n) = rows n that have complete residue coverage, i.e., A066981(n) = n and A223487(n) = 0.

Example

Row 1 contains 0 by convention.
Row 2 contains (0, 1) since the Lucas sequence contains both even and odd numbers.
Row 5 contains (1, 2, 3, 4) since the Lucas numbers mod 5 is {2,1,3,4,2,1} repeated; we are missing the residue 0.
Table begins as shown below, with residue k shown arranged in columns.
n    k (mod n)
--------------
1:   0
2:   0  1
3:   0  1  2
4:   0  1  2  3
5:      1  2  3  4
6:   0  1  2  3  4  5
7:   0  1  2  3  4  5  6
8:      1  2  3  4  5     7
9:   0  1  2  3  4  5  6  7  8
10:     1  2  3  4     6  7  8  9
11:  0  1  2  3  4        7       10
12:     1  2  3  4  5  6  7  8    10 11
13:     1  2  3  4  5  6  7  8  9 10 11 12
14:  0  1  2  3  4  5  6  7  8  9 10 11 12 13
15:     1  2  3  4        7          11       14
16:     1  2  3  4  5     7     9    11 12 13    15
...

Mathematica

{Most@ #, #} &[Range[0, 1]]~Join~Array[Block[{w = {2, 1}}, Do[If[SequenceCount[w, {2, 1}] == 1, AppendTo[w, Mod[Total@ w[[-2 ;; -1]], #]], Break[]], {i, 2, Infinity}]; Union@ w] &, 12, 3] // Flatten

Crossrefs

Cf. A000032, A066981, A106291, A223487. Analogous to A189768.

Sequence in context: A363612 A309816 A071486 * A046695 A071433 A110657

Adjacent sequences: A336681 A336682 A336683 * A336685 A336686 A336687

Keyword

nonn,tabf,easy

Author

Michael De Vlieger, Oct 07 2020

Status

approved