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A212894 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)=1.

%I A212894 #18 Oct 27 2024 04:16:18
%S A212894 0,2,22,92,246,520,950,1572,2422,3536,4950,6700,8822,11352,14326,
%T A212894 17780,21750,26272,31382,37116,43510,50600,58422,67012,76406,86640,
%U A212894 97750,109772,122742,136696,151670,167700,184822,203072,222486,243100
%N A212894 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)=1.
%C A212894 The gapsizes are |w-x|, |x-y|, |y-z|.  Every term is even. For a guide to related sequences, see A211795.
%H A212894 Vincenzo Librandi, <a href="/A212894/b212894.txt">Table of n, a(n) for n = 0..1000</a>
%H A212894 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A212894 a(n) = 2*(n-1)*(3*n^2-3*n+5) with n>1, a(0)=0, a(1)=2.
%F A212894 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>=6.
%F A212894 G.f.: f(x)/g(x), where f(x)=2*(x+7*x^2+8*x^3+x^4+x^5) and g(x)=(1-x)^4.
%t A212894 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212894 (Do[If[Min[Abs[w - x], Abs[x - y], Abs[y - z]] == 1, s = s + 1],
%t A212894 {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t A212894 m = Map[t[#] &, Range[0, 40]]   (* A212894 *)
%t A212894 m/2 (* integers *)
%t A212894 CoefficientList[Series[2*(x+7*x^2+8*x^3+x^4+x^5) /(1-x)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 04 2012 *)
%o A212894 (Magma) I:=[0, 2, 22, 92, 246, 520]; [n le 6 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 A212894 Cf. A211795.
%K A212894 nonn,easy
%O A212894 0,2
%A A212894 _Clark Kimberling_, May 30 2012