# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a349545 Showing 1-1 of 1 %I A349545 #27 Jan 07 2024 17:17:22 %S A349545 1,1,1,4,1,6,64,4,6,168,4096,64,24,168,20160,1048576,4096,384,672, %T A349545 20160,9999360,1073741824,1048576,24576,10752,80640,9999360, %U A349545 20158709760,4398046511104,1073741824,6291456,688128,1290240,39997440,20158709760,163849992929280 %N A349545 Triangular array read by rows: T(n,k) = A002884(k)*2^((n-k)(n-k-1)), n >= 0, 0 <= k <= n. %C A349545 For A,B in the set of n X n matrices over GF(2) let A ~ B iff A^j = B^k for some positive j,k. Then ~ is an equivalence relation. There is exactly one idempotent matrix in each equivalence class. Let E be an idempotent matrix of rank k. Then T(n,k) is the size of the class containing E. %C A349545 The classes in the equivalence relation described above are called the torsion classes corresponding to the idempotent E. - _Geoffrey Critzer_, Oct 02 2022 %H A349545 Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50) %H A349545 Encyclopedia of Mathematics, Periodic semigroup %e A349545 Triangle begins: %e A349545 1; %e A349545 1, 1; %e A349545 4, 1, 6; %e A349545 64, 4, 6, 168; %e A349545 4096, 64, 24, 168, 20160; %e A349545 1048576, 4096, 384, 672, 20160, 9999360; %e A349545 ... %e A349545 T(3,1)=4 because we have: { I = {{0, 0, 0}, {0, 0, 0}, {0, 0, 1}}, %e A349545 A= {{0, 0, 0}, {1, 0, 0}, {0, 0, 1}}, B= {{0, 1, 0}, {0, 0, 0}, {0, 0, 1}}, %e A349545 C= {{1, 1, 0}, {1, 1, 0}, {0, 0, 1}} } where I is idempotent of rank 1 and A^2=B^2=C^2=I. %t A349545 q = 2; nn = 7;Table[Table[Product[q^d - q^i, {i, 0, d - 1}] q^((n - d) (n - d - 1)), {d, 0,n}], {n, 0, nn}] // Grid %o A349545 (PARI) \\ here b(n) is A002884(n). %o A349545 b(n) = {prod(i=2, n, 2^i-1)<