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A224673 Number of (n+1) X 6 0..2 matrices with each 2 X 2 subblock idempotent.

%I A224673 #12 Jul 23 2025 05:17:45
%S A224673 115,191,257,381,542,793,1166,1746,2650,4080,6355,9996,15843,25257,
%T A224673 40439,64951,104556,168579,272108,439556,710424,1148626,1857577,
%U A224673 3004606,4860457,7863203,12721661,20582721,33302090,53882365,87181850,141061446
%N A224673 Number of (n+1) X 6 0..2 matrices with each 2 X 2 subblock idempotent.
%C A224673 Column 5 of A224676.
%H A224673 R. H. Hardin, <a href="/A224673/b224673.txt">Table of n, a(n) for n = 1..210</a>
%F A224673 Empirical: a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5) for n > 6.
%F A224673 Conjectures from _Colin Barker_, Feb 17 2018: (Start)
%F A224673 G.f.: x*(115 - 269*x + 68*x^2 + 193*x^3 - 118*x^4 + 6*x^5) / ((1 - x)^3*(1 - x - x^2)).
%F A224673 a(n) = 34 + (2^(-1-n)*((1-sqrt(5))^n*(-11+53*sqrt(5)) + (1+sqrt(5))^n*(11+53*sqrt(5)))) / sqrt(5) + 14*(1+n) + (5/2)*(1 + n)*(2+n) for n>1.
%F A224673 (End)
%e A224673 Some solutions for n=3:
%e A224673   1 0 0 0 0 0    1 0 0 0 0 0    0 0 1 0 0 0    1 0 0 0 0 0
%e A224673   0 0 0 0 0 0    1 0 0 0 0 0    0 0 1 0 0 0    1 0 0 0 0 0
%e A224673   0 0 0 0 0 0    1 0 0 0 0 0    0 0 1 0 0 0    0 0 0 0 0 0
%e A224673   2 1 1 1 1 1    2 0 0 1 1 1    0 0 1 0 0 0    2 1 1 1 1 1
%K A224673 nonn
%O A224673 1,1
%A A224673 _R. H. Hardin_, Apr 14 2013