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A355008
Triangle T(n, k), n > 0, k = 1..n, read by rows; each row contains the smallest distinct positive integers (in ascending order) such that a term of row n times a term of row n+1 does not appear in row n+2.
1
1, 1, 2, 3, 4, 5, 1, 2, 7, 9, 1, 2, 7, 9, 11, 3, 5, 6, 8, 10, 12, 1, 2, 4, 7, 9, 11, 13, 1, 2, 4, 7, 9, 11, 13, 14, 3, 5, 6, 10, 12, 15, 17, 19, 20, 1, 2, 4, 7, 8, 9, 11, 13, 14, 16, 1, 2, 4, 7, 8, 9, 11, 13, 14, 16, 18, 3, 5, 6, 10, 12, 15, 17, 19, 20, 21, 23, 24
OFFSET
1,3
COMMENTS
Conjecture:
- rows with indexes of the form 3*k converge to some sequence R:
R = 3, 5, 6, 10, 12, 15, 17, 19, 20, 21, 23, 24, 27, 29, 30, 33, 34, ...
- the other rows converge to some other sequence S:
S = 1, 2, 4, 7, 8, 9, 11, 13, 14, 16, 18, 22, 25, 26, 28, 31, 32, 36, ...
- R and S partition the positive integers,
- for any i, j, k:
R(i) <> S(j) * S(k),
S(i) <> R(k) * S(k).
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10011 (rows for n = 1..141 flattened)
EXAMPLE
Triangle T(n, k) begins:
1,
1, 2,
3, 4, 5,
1, 2, 7, 9,
1, 2, 7, 9, 11,
3, 5, 6, 8, 10, 12,
1, 2, 4, 7, 9, 11, 13,
1, 2, 4, 7, 9, 11, 13, 14,
3, 5, 6, 10, 12, 15, 17, 19, 20,
1, 2, 4, 7, 8, 9, 11, 13, 14, 16,
1, 2, 4, 7, 8, 9, 11, 13, 14, 16, 18,
3, 5, 6, 10, 12, 15, 17, 19, 20, 21, 23, 24,
...
PROG
(PARI) { rrr=rr=[]; for (n=1, 12, x=setbinop((i, j)->i*j, rrr, rr); r=vector(n); k=0; for (v=1, oo, if (!setsearch(x, v), print1 (r[k++]=v", "); if (k==n, break))); [rrr, rr]=[rr, r]) }
CROSSREFS
Cf. A135018.
Sequence in context: A193042 A327463 A279478 * A050269 A097151 A306620
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Jun 15 2022
STATUS
approved