A122554 - OEIS

%I #30 Jan 04 2024 15:11:36

%S 1,3,6,10,15,23,35,54,84,132,209,333,533,856,1378,2222,3587,5795,9367,

%T 15146,24496,39624,64101,103705,167785,271468,439230,710674,1149879,

%U 1860527,3010379,4870878,7881228,12752076,20633273,33385317,54018557,87403840,141422362

%N 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 number of elements in S(n).

%C If the set mapping has x -> x,2x,x^2 is used instead of x -> x,x+2,2x, the corresponding sequence consists of the Fibonacci numbers 1,2,3,5,8,...

%C Apparently a(n)= 3*a(n-1) -2*a(n-2) -a(n-3) +a(n-4) for n>6, equivalent to a(n)=A000032(n)+n-1 for n>2. - _R. J. Mathar_, Nov 18 2009

%H Chai Wah Wu, <a href="/A122554/b122554.txt">Table of n, a(n) for n = 1..43</a>

%F Empirical g.f.: -x*(x^5-x^4-x^3-x^2+1) / ((x-1)^2*(x^2+x-1)). - _Colin Barker_, Nov 06 2014

%e Under the indicated set mapping we have {1} -> {1,2,3} -> {1,2,3,4,5,6} -> {1,2,3,4,5,6,7,8,10,12}, ..., so a(2)=3, a(3)=6, a(4)=10, etc.

%t Do[ Print@ Length@ Nest[ Union@ Flatten[ # /. a_Integer -> {a, 2a, a + 2}] &, {1}, n], {n, 0, 32}] (* _Robert G. Wilson v_, Sep 27 2006 *)

%o (Python)

%o from sympy import chain, islice

%o def A122554_gen(): # generator of terms

%o     s = {1}

%o     while True:

%o         yield len(s)

%o         s = set(chain.from_iterable((x,x+2,2*x) for x in s))

%o A122554_list = list(islice(A122554_gen(),24)) # _Chai Wah Wu_, Jan 12 2022

%K nonn

%O 1,2

%A _John W. Layman_, Sep 20 2006

%E a(17)-a(33) from _Robert G. Wilson v_, Sep 27 2006

%E a(34)-a(36) from _Jinyuan Wang_, Apr 14 2020

%E a(37)-a(39) from _Chai Wah Wu_, Jan 12 2022