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

1,2

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

Links

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

Sequence in context: A262927 A063542 A294413 * A111734 A372111 A117457

Adjacent sequences: A122551 A122552 A122553 * A122555 A122556 A122557

Keyword

nonn

Author

John W. Layman, Sep 20 2006

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

Status

approved