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A212899 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)>2.

%I A212899 #16 Jun 01 2025 15:01:25
%S A212899 0,0,0,2,16,62,186,456,962,1818,3162,5156,7986,11862,17018,23712,
%T A212899 32226,42866,55962,71868,90962,113646,140346,171512,207618,249162,
%U A212899 296666,350676,411762,480518,557562,643536,739106,844962,961818
%N A212899 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)>2.
%C A212899 The gapsizes are |w-x|, |x-y|, |y-z|.  Every term is even. a(n)+A212898(n)=(n+1)^4.
%C A212899 For a guide to related sequences, see A211795.
%H A212899 Harvey P. Dale, <a href="/A212899/b212899.txt">Table of n, a(n) for n = 0..1000</a>
%H A212899 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F A212899 a(n)=5*a(n-1)+10*a(n-2)-10*a(n-3)+5*a(n-4)-a(n-5) for n>=9.
%F A212899 G.f.: (2*x^3 + 6*x^4 + 2*x^5 + 16*x^6 - 4*x^7 + 2*x^9 )/(1 - 5*x + 10*x^2 - 10*x^3 + 5*x^4 - x^5)
%t A212899 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212899 (Do[If[Min[Abs[w - x], Abs[x - y], Abs[y - z]] > 2, s = s + 1],
%t A212899 {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t A212899 m = Map[t[#] &, Range[0, 40]]   (* A212899 *)
%t A212899 m/2  (* integers *)
%t A212899 LinearRecurrence[{5,-10,10,-5,1},{0,0,0,2,16,62,186,456,962,1818},40] (* _Harvey P. Dale_, Jun 01 2025 *)
%Y A212899 Cf. A211795.
%K A212899 nonn,easy
%O A212899 0,4
%A A212899 _Clark Kimberling_, May 31 2012