cp's OEIS Frontend

The canonical representation of a given Boolean function f of n variables by its Algebraic Normal Form (ANF) is a multivariate polynomial of the form f(x_1, x_2, ..., x_n)= a + M_1 + M_2 + ... + M_s, where a is the Boolean constant 0 or 1, the + sign denotes sum modulo 2 (XOR) between different monomials, and 0 <= s < 2^n. The monomial M_i is a conjunction of some of these variables, for i= 1, 2, ..., s. The algebraic degree of a monomial is the number of variables in the monomial and the algebraic degree of a Boolean function is the maximum degree of the monomials. There are binomial(n, k) monomials of k variables, for k= 0, 1, ..., n. The case k= 0 means that no one of the variables is chosen to form a monomial. It corresponds to the Boolean constant 1, which is considered as a monomial. Its algebraic degree is equal to 0 and so T(n,0)= 1, for n= 0, 1, ... The zero constant is not considered as a monomial. It corresponds to the empty set - the case when no one of the possible monomials is chosen to form an ANF. Its algebraic degree is defined usually as -infinity. That is why the zero constant is not represented in the triangle.

The set of all Boolean functions of n variables can be partitioned into subsets in accordance with their algebraic degrees. The n-th row of the triangle represents the cardinalities of these subsets. Thus the sum of all numbers in the n-th row is the number of all Boolean functions of n variables - 1 (because of the zero constant), i.e., 2^(2^n)-1.

Equivalently, the number of nonempty sets of subsets of an n-set such that the maximum cardinality of the subsets is k. - Andrew Howroyd, Sep 23 2018

Analog of A051179 replacing 2 with e.

A051179(n) is the least number k such that a(k) = n.

The asymptotic density of occurrences of 0 is 1/2.

The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/2^(2^k) = (1/2) * A007404 = 0.4082107545... .

The multiplicity of a divisor d (not necessarily a prime) of n is defined in A169594 (see also the first formula).

A000244(n) is the least number k such that a(k) = n.

The asymptotic density of occurrences of 0 is 1/2.

The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/(2^(2^k)+1) = (1/2) * A051158 = 0.2980315860... .

Row n has 2^(n+1)-2 terms. In row n first nonzero term is T(n,n)=2^n and last nonzero term is T(n,2^(n+1)-2)=1. Row sums yield A051179. Column sums yield A106376.

There are an infinite number of sequences {a(k)}, with different values for a(1) and a(2) (a(1) must be 0 or 1; a(2) can be anything), where Product_{k=1..n} (Sum_{j=1..k} a(j)) = Sum_{k=1..n} Product_{j=1..k} a(j), for all positive integers n. Setting a(1) to 1 and a(2) to 2 results in the sequence here.

All sequences (not necessarily integer sequences) with a(1) = 0 trivially have the property in the sequence name because each product is zero. For a general sequence in this family with a(1) = 1 and a(2) any integer, then a(3) = -a(2)^2 - a(2) and, for n >= 4, a(n) = -a(2)^(2^(n-3))*(a(2)^(2^(n-3))-1), so that all terms after a(2) are negatives of oblong (or promic) numbers (A002378). - Rick L. Shepherd, Aug 10 2014

A051179 is a bisection of this sequence.

Sequence is finite with only 7 values. With the exception of m = 5, the others are products of the first k Fermat primes; i.e., products of A019434 and matching the initial terms of A051179. With the exception of m = 5, sequence resembles A250405.

Numbers m such that m and m+1 are in A003401

Each of m and m+1 must be a power of 2 times a product of Fermat primes.

Apart from term 5, odd terms are of the form 2^2^k - 1 for k in 0...5.

Even terms are exactly numbers of the form 2^2^k such that 2^2^k + 1 is a Fermat prime (A019434).

The sequence is thus infinite iff A019434 is infinite, i.e., iff there are infinitely many Fermat primes.