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A123212
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Let S(1) = {1} and, for n > 1, let S(n) be the smallest set containing x, 2x and x^2 for each element x in S(n-1). a(n) is the sum of the elements in S(n).
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1
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1, 3, 7, 31, 383, 71679, 4313284607, 18447026747376402431, 340282367000167840050178713574329810943, 115792089237316195429848086745536112650120661123018741407845920610578123980799
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MAPLE
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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)):
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MATHEMATICA
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S[n_] := S[n] = If[n == 1, {1}, {#, 2#, #^2}& /@ S[n-1] // Flatten // Union];
a[n_] := S[n] // Total;
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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