A122554 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).

1, 3, 6, 10, 15, 23, 35, 54, 84, 132, 209, 333, 533, 856, 1378, 2222, 3587, 5795, 9367, 15146, 24496, 39624, 64101, 103705, 167785, 271468, 439230, 710674, 1149879, 1860527, 3010379, 4870878, 7881228, 12752076, 20633273, 33385317, 54018557, 87403840, 141422362

Offset: 1

Author

John W. Layman, Sep 20 2006

Keywords

• nonn

Comments

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,...

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

Examples

			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.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..43

Programs

Mathematica

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

Python

from sympy import chain, islice
def A122554_gen(): # generator of terms
    s = {1}
    while True:
        yield len(s)
        s = set(chain.from_iterable((x,x+2,2*x) for x in s))
A122554_list = list(islice(A122554_gen(),24)) # Chai Wah Wu, Jan 12 2022

Formula

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

Extensions

a(17)-a(33) from Robert G. Wilson v, Sep 27 2006

a(34)-a(36) from Jinyuan Wang, Apr 14 2020

a(37)-a(39) from Chai Wah Wu, Jan 12 2022