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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.

%I A386214 #18 Jul 28 2025 03:21:48
%S A386214 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,
%T A386214 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,
%U A386214 4,3,2,1,233,22,14,12,12,7,6,5,4,3,2,1
%N 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.
%H A386214 Clark Kimberling  (proposer), P. Bruckman and P. L. Mana (solvers), <a href="https://www.fq.math.ca/Scanned/28-4/elementary28-4.pdf">Problem B-657, Disjoint Increasing Sequences</a>, Fibonacci Quarterly, 30 (1990), 375.
%e A386214 Corner of the array:
%e A386214   1   2   3   5   8  13  21  34  55   89  144  233  377  610  987
%e A386214   1   2   3   4   6   7   8  11  14   18   22   29   36   47   58
%e A386214   1   2   3   4   5   8   9  10  12   14   15   18   23   27   28
%e A386214   1   2   3   4   5   6  10  11  12   15   17   18   20   22   24
%e A386214   1   2   3   4   5   6   7  12  13   14   18   20   21   24   26
%e A386214   1   2   3   4   5   6   7   8  14   15   16   21   23   24   28
%e A386214   1   2   3   4   5   6   7   8   9   16   17   18   24   26   27
%e A386214 (row 3) is the union, in increasing order, of these 3 disjoint sequences:
%e A386214   (1, 4, 5, 9, 14, 23, 37, 60, 97, 157, ...);
%e A386214   (2, 8, 10, 18, 28, 46, 74, 120, 194, ...);
%e A386214   (3, 12, 15, 27, 42, 69, 111, 180, 291, ...).
%e A386214 All three sequences are multiples of the first.
%t A386214 f[n_] := Fibonacci[n];
%t A386214 t[m_] := Table[k ((m+1)*f[n] + f[n - 1]), {k, 1, m}, {n, 0, 30}];
%t A386214 tt = Table[Sort[Flatten[t[m]]], {m, 1, 14}];
%t A386214 Column[tt]  (* array *)
%t A386214 u[n_, k_] := tt[[n]][[k]];
%t A386214 Table[u[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
%Y A386214 Cf. A000045 (row 1), A127218 (row 2, except for initial terms), A000027 (limiting row), A039834.
%K A386214 nonn,tabl
%O A386214 1,2
%A A386214 _Clark Kimberling_, Jul 15 2025