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

Author

Tilman Piesk, Jul 24 2012

Keywords

• nonn
• tabl

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, here shown with k starting from 0, so that n-ary x_k = T(n, n-k-1). As numbers in the diagonals on the right have the same bit pattern, this corresponds to the infinitary definition of x_k as a binary fraction 1/A000215(k) = 1/(2^2^k + 1):

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)

Crossrefs

A001317, A089633, A051179 (left diagonal)

Programs

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<A211344_list = list(islice(A211344_gen(),20)) # Chai Wah Wu, May 03 2023

Python

def arity_and_atom_to_integer(arity, atom):
    result = 0
    max_place = (1 << arity) - (1 << atom) - 1
    for exponent in range(max_place + 1):
        if not bool(~max_place & max_place - exponent):
            place_value = 1 << exponent
            result += place_value
    return result
def A211344(n, k):
    return arity_and_atom_to_integer(n, n-k-1) # Tilman Piesk, Jan 25 2025

Formula

a = A001317( A089633 )