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, 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
Examples
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.
Links
- Clark Kimberling (proposer), P. Bruckman and P. L. Mana (solvers), Problem B-657, Disjoint Increasing Sequences, Fibonacci Quarterly, 30 (1990), 375.
Crossrefs
Programs
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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 *)