A350605 Irregular triangle read by rows: row n lists the elements of the set S_n in increasing order, where S_1 = {1}, and S_{n+1} is the set {k, 2*k+1, 3*k+1: k in S_n}.

1, 1, 3, 4, 1, 3, 4, 7, 9, 10, 13, 1, 3, 4, 7, 9, 10, 13, 15, 19, 21, 22, 27, 28, 31, 40, 1, 3, 4, 7, 9, 10, 13, 15, 19, 21, 22, 27, 28, 31, 39, 40, 43, 45, 46, 55, 57, 58, 63, 64, 67, 81, 82, 85, 94, 121, 1, 3, 4, 7, 9, 10, 13, 15, 19, 21, 22, 27, 28, 31, 39, 40, 43, 45, 46, 55, 57, 58, 63, 64, 67, 79, 81, 82, 85, 87, 91, 93, 94, 111, 115, 117, 118, 121, 127, 129, 130, 135, 136, 139, 163, 165, 166, 171, 172, 175, 189, 190, 193, 202, 243, 244, 247, 256, 283, 364

Offset

1,3

Comments

Row n has A350606(n) elements.

The rows converge to A002977.

Links

Alois P. Heinz, Rows n = 1..15, flattened

David A. Klarner and Richard Rado, Linear combinations of sets of consecutive integers, The American Mathematical Monthly, Vol. 80, No. 9 (1973), pp. 985-989.

Example

The first few sets S_n are:
[1],
[1, 3, 4],
[1, 3, 4, 7, 9, 10, 13],
[1, 3, 4, 7, 9, 10, 13, 15, 19, 21, 22, 27, 28, 31, 40],
[1, 3, 4, 7, 9, 10, 13, 15, 19, 21, 22, 27, 28, 31, 39, 40, 43, 45, 46, 55, 57, 58, 63, 64, 67, 81, 82, 85, 94, 121],
...

Maple

T:= proc(n) option remember; `if`(n=1, 1, sort(
     [map(k-> [k, 2*k+1, 3*k+1][], {T(n-1)})[]])[])
    end:
seq(T(n), n=1..6);  # Alois P. Heinz, Jan 12 2022

Mathematica

T[n_] := T[n] = If[n==1, {1}, {#, 2#+1, 3#+1}& /@ T[n-1] // Flatten //
     Union];
Table[T[n], {n, 1, 6}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)

Prog

(Python)
from itertools import chain, islice
def A350605_gen(): # generator of terms
    s = {1}
    while True:
        yield from sorted(s)
        s = set(chain.from_iterable((x, 2*x+1, 3*x+1) for x in s))
A350605_list = list(islice(A350605_gen(), 30)) # Chai Wah Wu, Jan 12 2022

Crossrefs

Cf. A002977, A350604, A350606.

Sequence in context: A117378 A278518 A088197 * A087517 A128529 A205547

Adjacent sequences: A350602 A350603 A350604 * A350606 A350607 A350608

Keyword

nonn,look,tabf

Author

N. J. A. Sloane, Jan 12 2022

Status

approved