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Number of (w,x,y) with all terms in {0,...,n} and the numbers w,x,y,|w-x|,|x-y| distinct.
2

%I #14 Feb 19 2024 10:27:26

%S 0,0,0,0,0,12,38,92,160,286,422,632,870,1194,1542,2010,2502,3126,3788,

%T 4598,5446,6472,7532,8786,10092,11604,13164,14964,16812,18912,21074,

%U 23504,25996,28786,31634,34796,38034,41598,45234,49230,53298

%N Number of (w,x,y) with all terms in {0,...,n} and the numbers w,x,y,|w-x|,|x-y| distinct.

%C For a guide to related sequences, see A212959.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-2,0,0,1,1,-1).

%F a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10).

%F G.f.: (12*x^5 + 26*x^6 + 42*x^7 + 30*x^8 + 34*x^9)/(1 - x - x^2 + 2*x^5 - x^8 - x^9 + x^10).

%F a(n) = (n+1)^3 - A213491(n).

%t t = Compile[{{n, _Integer}}, Module[{s = 0},

%t (Do[If[Length[Union[{w, x, y, Abs[w - x],

%t Abs[x - y]}]] == 5,

%t s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];

%t m = Map[t[#] &, Range[0, 60]] (* this sequence *)

%t LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {0, 0, 0, 0, 0, 12, 38, 92, 160, 286}, 60]

%t m/2 (* integers *)

%Y Cf. A212959, A213491.

%K nonn,easy

%O 0,6

%A _Clark Kimberling_, Jun 13 2012