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A214596 Number of 3X3X3 triangular 0..n arrays with every horizontal row having the same average value.

%I A214596 #12 Aug 21 2025 10:00:32
%S A214596 2,23,62,157,312,601,986,1619,2426,3589,5028,6997,9314,12371,15914,
%T A214596 20377,25488,31789,38846,47399,56906,68161,80592,95137,111002,129383,
%U A214596 149342,172141,196800,224761,254762,288563,324722,365077,408132,455941,506666
%N A214596 Number of 3X3X3 triangular 0..n arrays with every horizontal row having the same average value.
%H A214596 R. H. Hardin, <a href="/A214596/b214596.txt">Table of n, a(n) for n = 1..210</a>
%F A214596 Empirical: a(n) = 3*a(n-2) +2*a(n-3) -3*a(n-4) -6*a(n-5) +6*a(n-7) +3*a(n-8) -2*a(n-9) -3*a(n-10) +a(n-12).
%F A214596 Empirical: G.f. ( -x*(2+23*x+56*x^2+84*x^3+86*x^4+87*x^5+60*x^6+23*x^7-2*x^9+x^11) ) / ( (1+x+x^2)^2*(1+x)^3*(x-1)^5 ). - _R. J. Mathar_, Jul 23 2012
%e A214596 Some solutions for n=4
%e A214596 ....3......2......1......1......2......2......3......2......3......2......2
%e A214596 ...2.4....1.3....1.1....2.0....3.1....0.4....3.3....1.3....4.2....4.0....0.4
%e A214596 ..2.4.3..3.2.1..1.0.2..1.1.1..3.1.2..2.0.4..3.2.4..3.3.0..2.3.4..1.2.3..0.3.3
%Y A214596 Row 3 of A214595.
%K A214596 nonn,changed
%O A214596 1,1
%A A214596 _R. H. Hardin_, Jul 22 2012