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A354236
A(n,k) is the n-th number m such that the Collatz (or 3x+1) trajectory starting at m contains exactly k odd integers; square array A(n,k), n>=1, k>=1, read by antidiagonals.
14
1, 5, 2, 3, 10, 4, 17, 6, 20, 8, 11, 34, 12, 21, 16, 7, 22, 35, 13, 40, 32, 9, 14, 23, 68, 24, 42, 64, 25, 18, 15, 44, 69, 26, 80, 128, 33, 49, 19, 28, 45, 70, 48, 84, 256, 43, 65, 50, 36, 29, 46, 75, 52, 85, 512, 57, 86, 66, 51, 37, 30, 88, 136, 53, 160, 1024
OFFSET
1,2
FORMULA
A078719(A(n,k)) = k.
EXAMPLE
Square array A(n,k) begins:
1, 5, 3, 17, 11, 7, 9, 25, 33, 43, ...
2, 10, 6, 34, 22, 14, 18, 49, 65, 86, ...
4, 20, 12, 35, 23, 15, 19, 50, 66, 87, ...
8, 21, 13, 68, 44, 28, 36, 51, 67, 89, ...
16, 40, 24, 69, 45, 29, 37, 98, 130, 172, ...
32, 42, 26, 70, 46, 30, 38, 99, 131, 173, ...
64, 80, 48, 75, 88, 56, 72, 100, 132, 174, ...
128, 84, 52, 136, 90, 58, 74, 101, 133, 177, ...
256, 85, 53, 138, 92, 60, 76, 102, 134, 178, ...
512, 160, 96, 140, 93, 61, 77, 196, 260, 179, ...
MAPLE
b:= proc(n) option remember; irem(n, 2, 'r')+
`if`(n=1, 0, b(`if`(n::odd, 3*n+1, r)))
end:
A:= proc() local h, p, q; p, q:= proc() [] end, 0;
proc(n, k)
if k=1 then return 2^(n-1) fi;
while nops(p(k))<n do q:= q+1;
h:= b(q);
p(h):= [p(h)[], q]
od; p(k)[n]
end
end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
MATHEMATICA
b[n_] := b[n] = Module[{q, r}, {q, r} = QuotientRemainder[n, 2]; r +
If[n == 1, 0, b[If[OddQ[n], 3*n + 1, q]]]];
A = Module[{h, p, q}, p[_] = {}; q = 0;
Function[{n, k}, If[k == 1, 2^(n - 1)];
While[Length[p[k]] < n, q = q + 1;
h = b[q];
p[h] = Append[p[h], q]];
p[k][[n]]]];
Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Jun 02 2022, after Alois P. Heinz *)
CROSSREFS
Row n=1 gives A092893(k-1).
Sequence in context: A261026 A160080 A026247 * A348779 A226693 A306300
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 20 2022
STATUS
approved