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A224669 Number of (n+1) X 2 0..2 matrices with each 2 X 2 subblock idempotent.

%I A224669 #8 Feb 17 2018 05:32:16
%S A224669 12,25,41,70,115,189,308,501,813,1318,2135,3457,5596,9057,14657,23718,
%T A224669 38379,62101,100484,162589,263077,425670,688751,1114425,1803180,
%U A224669 2917609,4720793,7638406,12359203,19997613,32356820,52354437,84711261,137065702
%N A224669 Number of (n+1) X 2 0..2 matrices with each 2 X 2 subblock idempotent.
%C A224669 Column 1 of A224676.
%H A224669 R. H. Hardin, <a href="/A224669/b224669.txt">Table of n, a(n) for n = 1..210</a>
%F A224669 Empirical: a(n) = 2*a(n-1) -a(n-3).
%F A224669 Conjectures from _Colin Barker_, Feb 17 2018: (Start)
%F A224669 G.f.: x*(12 + x - 9*x^2) / ((1 - x)*(1 - x - x^2)).
%F A224669 a(n) = -4 + (2^(-1-n)*((1-sqrt(5))^n*(-19+13*sqrt(5)) + (1+sqrt(5))^n*(19+13*sqrt(5)))) / sqrt(5).
%F A224669 (End)
%e A224669 Some solutions for n=3:
%e A224669 ..1..0....0..0....0..0....1..0....1..2....1..2....0..0....1..2....1..1....0..0
%e A224669 ..1..0....0..0....1..1....0..1....0..0....0..0....0..1....0..0....0..0....1..1
%e A224669 ..0..1....0..0....0..0....0..1....0..0....0..0....0..1....0..0....1..1....0..0
%e A224669 ..0..1....2..1....0..0....0..1....1..1....0..0....0..1....2..1....0..0....0..1
%K A224669 nonn
%O A224669 1,1
%A A224669 _R. H. Hardin_, Apr 14 2013