OFFSET
1,2
COMMENTS
If we take the cardinality of the set S(n) instead of the sum, we get the Fibonacci numbers 1,2,3,5,8,13,21,34,... If the set mapping uses x -> x, 2x and 3x instead of x -> x, 2x, and x^2, the corresponding sequence consists of the Stirling numbers of the second kind: 1, 6, 25, 90, 301, 966, 3025, ... (A000392).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..13
EXAMPLE
Under the indicated set mapping we have {1} -> {1,2} -> {1,2,4} -> {1,2,4,8,16}, giving the sums a(1)=1, a(2)=3, a(3)=7, a(4)=31, etc.
MAPLE
s:= proc(n) option remember; `if`(n=1, 1,
map(x-> [x, 2*x, x^2][], {s(n-1)})[])
end:
a:= n-> add(i, i=s(n)):
seq(a(n), n=1..10); # Alois P. Heinz, Jan 12 2022
MATHEMATICA
S[n_] := S[n] = If[n == 1, {1}, {#, 2#, #^2}& /@ S[n-1] // Flatten // Union];
a[n_] := S[n] // Total;
Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Apr 22 2022 *)
PROG
(Python)
from itertools import chain, islice
def A123212_gen(): # generator of terms
s = {1}
while True:
yield sum(s)
s = set(chain.from_iterable((x, 2*x, x**2) for x in s))
CROSSREFS
KEYWORD
nonn
AUTHOR
John W. Layman, Oct 05 2006
STATUS
approved