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A336684
Irregular triangle in which row n lists residues k found in the sequence Lucas(i) mod n.
1
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.
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
KEYWORD
nonn,tabf,easy
AUTHOR
Michael De Vlieger, Oct 07 2020
STATUS
approved