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%I #20 May 04 2023 02:16:29
%S 1,3,5,15,51,85,255,3855,13107,21845,65535,16711935,252645135,
%T 858993459,1431655765,4294967295,281470681808895,71777214294589695,
%U 1085102592571150095,3689348814741910323,6148914691236517205
%N Atomic Boolean functions interpreted as binary numbers.
%C Row n of the triangle shows the atoms among n-ary Boolean functions:
%C 1 01
%C 3 5 0011 0101
%C 15 51 85 00001111 00110011 01010101
%C 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.
%C 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:
%C 2-ary x_0=0101=5, 3-ary x_0=01010101=85, infinitary x_0=1/3=.010101...
%C 2-ary x_1=0011=3, 3-ary x_1=00110011=51, infinitary x_1=1/5=.001100110011...
%H Tilman Piesk, <a href="/A211344/b211344.txt">Table of n, a(n) for n = 0..65</a>
%H Tilman Piesk, <a href="http://commons.wikimedia.org/wiki/File:Atomic_Boolean_functions_in_Sierpinski_triangle.svg">Atomic Boolean functions in Sierpinski triangle</a> (Wikimedia Commons)
%F a = A001317( A089633 )
%o (MATLAB)
%o Seq = sym(zeros(55,1)) ;
%o Filledlines = 0 ;
%o for m=1:10
%o for n=1:m
%o Sum = sym(0) ;
%o for k=0:2^m-1
%o if mod( floor( k/2^(m-n) ) ,2) == 0
%o Sum = Sum + 2^sym(k) ;
%o end
%o end
%o Seq( Filledlines + n ) = Sum ;
%o end
%o Filledlines = Filledlines + m ;
%o end
%o (Python)
%o from itertools import count, islice
%o def A211344_gen(): # generator of terms
%o 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))
%o A211344_list = list(islice(A211344_gen(),20)) # _Chai Wah Wu_, May 03 2023
%Y A001317, A089633, A051179 (left diagonal)
%K nonn,tabl
%O 0,2
%A _Tilman Piesk_, Jul 24 2012
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