A211344 Atomic Boolean functions interpreted as binary numbers.

1, 3, 5, 15, 51, 85, 255, 3855, 13107, 21845, 65535, 16711935, 252645135, 858993459, 1431655765, 4294967295, 281470681808895, 71777214294589695, 1085102592571150095, 3689348814741910323, 6148914691236517205

Offset

0,2

Comments

Row n of the triangle shows the atoms among n-ary Boolean functions:

1 01

3 5 0011 0101

15 51 85 00001111 00110011 01010101

Often n-ary x_k = T(n,k), e.g. for 2-ary functions x_1=0011, x_2=0101 and for 3-ary functions x_1=00001111, x_2=00110011, x_3=01010101.

An easier generalized way is the enumeration from right to left (preferably from x_0) so that n-ary x_k = T(n,n-k). As numbers in the diagonals on the right have the same bit pattern this goes well together with the infinitary definition of atomic formulas as x_k = 1/A000215(k) = 1/(2^2^k+1) in binary:

2-ary x_0=0101=5, 3-ary x_0=01010101=85, infinitary x_0=1/3=.010101...

2-ary x_1=0011=3, 3-ary x_1=00110011=51, infinitary x_1=1/5=.001100110011...

Links

Tilman Piesk, Table of n, a(n) for n = 0..65

Tilman Piesk, Atomic Boolean functions in Sierpinski triangle (Wikimedia Commons)

Formula

a = A001317( A089633 )

Prog

(MATLAB)
Seq = sym(zeros(55, 1)) ;
Filledlines = 0 ;
for m=1:10
    for n=1:m
        Sum = sym(0) ;
        for k=0:2^m-1
            if mod(  floor( k/2^(m-n) )  , 2) == 0
               Sum = Sum + 2^sym(k) ;
            end
        end
        Seq( Filledlines + n ) = Sum ;
    end
    Filledlines = Filledlines + m ;
end
(Python)
from itertools import count, islice
def A211344_gen(): # generator of terms
    return (sum((bool(~(m:=(1<<t)-(1<<k)-1)&m-i)^1)<<i for i in range((1<<t)-(1<<k))) for t in count(1) for k in range(t-1, -1, -1))
A211344_list = list(islice(A211344_gen(), 20)) # Chai Wah Wu, May 03 2023

Crossrefs

A001317, A089633, A051179 (left diagonal)

Sequence in context: A103043 A018601 A190733 * A006394 A018650 A177814

Adjacent sequences: A211341 A211342 A211343 * A211345 A211346 A211347

Keyword

nonn,tabl

Author

Tilman Piesk, Jul 24 2012

Status

approved