VOOZH about

URL: https://oeis.org/A386214

⇱ A386214 - OEIS

login
A386214
Rectangular array, read by descending antidiagonals: (row m) consists of the union, in increasing order, of the numbers in the following set: {k*((m+1)*F(n) + F(n - 1)): k = 1..m, n>=0}, where F = A000045, the Fibonacci numbers, with F(-1)=1 as in A039834.
1
1, 2, 1, 3, 2, 1, 5, 3, 2, 1, 8, 4, 3, 2, 1, 13, 6, 4, 3, 2, 1, 21, 7, 5, 4, 3, 2, 1, 34, 8, 8, 5, 4, 3, 2, 1, 55, 11, 9, 6, 5, 4, 3, 2, 1, 89, 14, 10, 10, 6, 5, 4, 3, 2, 1, 144, 18, 12, 11, 7, 6, 5, 4, 3, 2, 1, 233, 22, 14, 12, 12, 7, 6, 5, 4, 3, 2, 1
OFFSET
1,2
LINKS
Clark Kimberling (proposer), P. Bruckman and P. L. Mana (solvers), Problem B-657, Disjoint Increasing Sequences, Fibonacci Quarterly, 30 (1990), 375.
EXAMPLE
Corner of the array:
1 2 3 5 8 13 21 34 55 89 144 233 377 610 987
1 2 3 4 6 7 8 11 14 18 22 29 36 47 58
1 2 3 4 5 8 9 10 12 14 15 18 23 27 28
1 2 3 4 5 6 10 11 12 15 17 18 20 22 24
1 2 3 4 5 6 7 12 13 14 18 20 21 24 26
1 2 3 4 5 6 7 8 14 15 16 21 23 24 28
1 2 3 4 5 6 7 8 9 16 17 18 24 26 27
(row 3) is the union, in increasing order, of these 3 disjoint sequences:
(1, 4, 5, 9, 14, 23, 37, 60, 97, 157, ...);
(2, 8, 10, 18, 28, 46, 74, 120, 194, ...);
(3, 12, 15, 27, 42, 69, 111, 180, 291, ...).
All three sequences are multiples of the first.
MATHEMATICA
f[n_] := Fibonacci[n];
t[m_] := Table[k ((m+1)*f[n] + f[n - 1]), {k, 1, m}, {n, 0, 30}];
tt = Table[Sort[Flatten[t[m]]], {m, 1, 14}];
Column[tt] (* array *)
u[n_, k_] := tt[[n]][[k]];
Table[u[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
CROSSREFS
Cf. A000045 (row 1), A127218 (row 2, except for initial terms), A000027 (limiting row), A039834.
Sequence in context: A251721 A251722 A383334 * A304100 A179314 A204927
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jul 15 2025
STATUS
approved