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A193698 Number of arrays of -4..4 integers x(1..n) with every x(i) in a subsequence of length 1, 2 or 3 with sum zero.

%I A193698 #7 Jul 22 2025 12:28:09
%S A193698 1,9,69,317,1741,9385,50035,268453,1438203,7705011,41284941,221198177,
%T A193698 1185157449,6349965025,34022467117,182289038899,976686732167,
%U A193698 5232991207579,28037851489661,150224045883455,804885636881145
%N A193698 Number of arrays of -4..4 integers x(1..n) with every x(i) in a subsequence of length 1, 2 or 3 with sum zero.
%C A193698 Column 4 of A193702
%H A193698 R. H. Hardin, <a href="/A193698/b193698.txt">Table of n, a(n) for n = 1..200</a>
%F A193698 Empirical: a(n) = 6*a(n-1) -6*a(n-2) +3*a(n-3) +33*a(n-4) +97*a(n-5) +56*a(n-6) +410*a(n-7) +879*a(n-8) +1783*a(n-9) +3145*a(n-10) +7753*a(n-11) +10683*a(n-12) +18787*a(n-13) +28728*a(n-14) +40680*a(n-15) +46937*a(n-16) +53760*a(n-17) +15971*a(n-18) -19648*a(n-19) -105105*a(n-20) -160613*a(n-21) -198846*a(n-22) -149920*a(n-23) -86466*a(n-24) +71880*a(n-25) +113537*a(n-26) +220479*a(n-27) +140242*a(n-28) +136646*a(n-29) +17320*a(n-30) -6999*a(n-31) -64171*a(n-32) -52874*a(n-33) -51993*a(n-34) -26792*a(n-35) -14383*a(n-36) -1773*a(n-37) +1763*a(n-38) +3249*a(n-39) +2387*a(n-40) +1554*a(n-41) +741*a(n-42) +312*a(n-43) +109*a(n-44) +35*a(n-45) +8*a(n-46) +a(n-47)
%e A193698 Some solutions for n=6
%e A193698 ..0....3....4...-3...-3...-2....2....2....3....3...-3...-1...-4....2....2...-1
%e A193698 ..2...-3...-3....2...-1....2...-2...-2...-4...-3....2...-3....2...-2...-2....1
%e A193698 ..0....2...-1....1....4...-3....1....1....1....3....1....4....2...-1....0...-2
%e A193698 .-2...-2...-4...-2...-4....1....1...-1....2...-1....0....0...-4....1....2....1
%e A193698 ..1....0....4....2....2...-1....4....0...-3...-3....0....0....1...-4...-2...-3
%e A193698 .-1....0...-4...-2....2....1...-4....0....0....4....0....0...-1....3....0....2
%K A193698 nonn
%O A193698 1,2
%A A193698 _R. H. Hardin_ Aug 02 2011