A213487 - OEIS

%I #11 Jun 19 2024 17:53:25

%S 1,5,15,37,77,138,223,338,489,679,911,1191,1525,1916,2367,2884,3473,

%T 4137,4879,5705,6621,7630,8735,9942,11257,12683,14223,15883,17669,

%U 19584,21631,23816,26145,28621,31247,34029,36973,40082,43359,46810

%N Number of (w,x,y) with all terms in {0,...,n} and |w-x|+|x-y|+|y-w| <= w+x+y.

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

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-7,8,-7,4,-1).

%F a(n) = 4*a(n-1)-7*a(n-2)+8*a(n-3)-7*a(n-4)+4*a(n-5)-a(n-6).

%F G.f.: (1 + x + 2*x^2 + 4*x^3 + x^4)/((1 - x)^4 (1 + x^2)).

%F a(n) = (n+1)^3 - A213486(n).

%t t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[w + x + y >= Abs[w - x] + Abs[x - y] + Abs[y - w], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];

%t m = Map[t[#] &, Range[0, 60]]   (* A213487 *)

%t LinearRecurrence[{4,-7,8,-7,4,-1},{1,5,15,37,77,138},50] (* _Harvey P. Dale_, Jun 19 2024 *)

%Y Cf. A212959, A213486.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Jun 13 2012