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A224670 Number of (n+1) X 3 0..2 matrices with each 2 X 2 subblock idempotent.

%I A224670 #9 Jul 23 2025 05:17:40
%S A224670 25,50,76,123,191,300,470,741,1173,1866,2980,4775,7671,12348,19906,
%T A224670 32125,51885,83846,135548,219191,354515,573460,927706,1500873,2428261,
%U A224670 3928790,6356680,10285071,16641323,26925936,43566770,70492185,114058401
%N A224670 Number of (n+1) X 3 0..2 matrices with each 2 X 2 subblock idempotent.
%C A224670 Column 2 of A224676.
%H A224670 R. H. Hardin, <a href="/A224670/b224670.txt">Table of n, a(n) for n = 1..210</a>
%F A224670 Empirical: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5).
%F A224670 Conjectures from _Colin Barker_, Feb 17 2018: (Start)
%F A224670 G.f.: x*(25 - 50*x + x^2 + 44*x^3 - 21*x^4) / ((1 - x)^3*(1 - x - x^2)).
%F A224670 a(n) = -2 + 2^(1-n)*sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n)) + 2*(1+n) + (1+n)*(2+n)/2.
%F A224670 (End)
%e A224670 Some solutions for n=3:
%e A224670 ..1..0..2....0..0..0....1..1..1....1..0..0....1..0..0....0..0..0....1..0..0
%e A224670 ..0..0..1....0..0..0....0..0..0....0..0..1....0..0..0....0..0..0....0..0..0
%e A224670 ..0..0..1....0..0..0....0..0..0....0..0..1....0..0..1....0..0..0....0..0..0
%e A224670 ..0..0..1....0..0..0....0..0..1....0..0..1....0..0..1....1..1..1....0..0..0
%K A224670 nonn
%O A224670 1,1
%A A224670 _R. H. Hardin_, Apr 14 2013