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A212895 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)=2.

%I A212895 #15 Sep 08 2022 08:46:02
%S A212895 0,0,2,14,58,168,376,716,1224,1936,2888,4116,5656,7544,9816,12508,
%T A212895 15656,19296,23464,28196,33528,39496,46136,53484,61576,70448,80136,
%U A212895 90676,102104,114456,127768,142076,157416,173824,191336,209988
%N A212895 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)=2.
%C A212895 The gapsizes are |w-x|, |x-y|, |y-z|.  Every term is even.
%C A212895 For a guide to related sequences, see A211795.
%H A212895 Vincenzo Librandi, <a href="/A212895/b212895.txt">Table of n, a(n) for n = 0..1000</a>
%H A212895 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A212895 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>=9.
%F A212895 G.f.: f(x)/g(x), where f(x)=(2 x^2)*(1 + 3 x + 7 x^2 + 6 x^3 - x^4 + x^5 + x^6) and g(x)=(1 - x)^4.
%t A212895 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212895 (Do[If[Min[Abs[w - x], Abs[x - y], Abs[y - z]] == 2, s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t A212895 m = Map[t[#] &, Range[0, 40]]   (* A212895 *)
%t A212895 m/2 (* integers *)
%t A212895 CoefficientList[Series[2 x^2*(1+3 x+7 x^2+6 x^3-x^4+x^5+x^6) /(1-x)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 04 2012 *)
%o A212895 (Magma) I:=[0, 0, 2, 14, 58, 168, 376, 716, 1224]; [n le 9 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jul 04 2012
%Y A212895 Cf. A211795.
%K A212895 nonn,easy
%O A212895 0,3
%A A212895 _Clark Kimberling_, May 31 2012