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A212896 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)<2.

%I A212896 #18 Aug 03 2025 21:05:15
%S A212896 1,16,79,240,551,1066,1839,2924,4375,6246,8591,11464,14919,19010,
%T A212896 23791,29316,35639,42814,50895,59936,69991,81114,93359,106780,121431,
%U A212896 137366,154639,173304,193415,215026,238191,262964,289399,317550
%N A212896 Number of (w,x,y,z) with all terms in {0,...,n} and (least gapsize)<2.
%C A212896 The gapsizes are |w-x|, |x-y|, |y-z|.
%C A212896 For a guide to related sequences, see A211795.
%H A212896 Vincenzo Librandi, <a href="/A212896/b212896.txt">Table of n, a(n) for n = 0..1000</a>
%H A212896 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A212896 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>=6.
%F A212896 G.f.: f(x)/g(x), where f(x) = 1+12*x+21*x^2+16*x^3+2*x^4+2*x^5 and g(x) = (1-x)^4.
%F A212896 a(n) = 9*n^3-6*n^2+20*n-9 with n>1, a(0)=1, a(1)=16. - _Bruno Berselli_, Jun 12 2012
%t A212896 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212896 (Do[If[Min[Abs[w - x], Abs[x - y], Abs[y - z]] <= 1, s = s + 1],
%t A212896 {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t A212896 m = Map[t[#] &, Range[0, 40]]   (* A212896 *)
%t A212896 CoefficientList[Series[(1+12*x+21*x^2+16*x^3+2*x^4+2*x^5) /(1-x)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 04 2012 *)
%o A212896 (Magma) I:=[1, 16, 79, 240, 551, 1066]; [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 A212896 Cf. A211795.
%K A212896 nonn,easy
%O A212896 0,2
%A A212896 _Clark Kimberling_, May 31 2012