OFFSET
0,2
COMMENTS
A Boolean relation matrix R is said to be convergent in its powers if in the sequence {R,R^2,R^3, ...} there is an m such that R^m = R^(m+1).
An idempotent Boolean relation matrix E is said to have a proper power primitive iff there is a convergent relation R with limit matrix E where R is not equal to E.
If an idempotent Boolean relation matrix E contains an identity matrix of order n-1 and (E-I_n)^2 = 0 then E has no proper power primitive. The converse is not true for n>=4. Consider {{1,0,1,0}, {0,1,0,1}, {0,0,0,0}, {0,0,0,0}}. The converse is erroneously stated and proved in Rosenblatt, Theorem 4.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..113
David Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research, National Bureau of Standards, Vol 67B No. 4 Oct-Dec 1963.
FORMULA
MAPLE
a:= n-> (n+1)*add(binomial(n, k)*(2^k-1)^(n-k), k=0..n):
seq(a(n), n=0..18); # Alois P. Heinz, Feb 18 2023
MATHEMATICA
nn = 16; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}]; Range[0, nn]! CoefficientList[Series[A[x] + x D[A[x], x], {x, 0, nn}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Feb 18 2023
EXTENSIONS
Corrected by Geoffrey Critzer, Feb 24 2023
STATUS
approved