A212894 - OEIS

%I #18 Oct 27 2024 04:16:18

%S 0,2,22,92,246,520,950,1572,2422,3536,4950,6700,8822,11352,14326,

%T 17780,21750,26272,31382,37116,43510,50600,58422,67012,76406,86640,

%U 97750,109772,122742,136696,151670,167700,184822,203072,222486,243100

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

%C The gapsizes are |w-x|, |x-y|, |y-z|.  Every term is even. For a guide to related sequences, see A211795.

%H Vincenzo Librandi, <a href="/A212894/b212894.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 2*(n-1)*(3*n^2-3*n+5) with n>1, a(0)=0, a(1)=2.

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>=6.

%F 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 t = Compile[{{n, _Integer}}, Module[{s = 0},

%t (Do[If[Min[Abs[w - x], Abs[x - y], Abs[y - z]] == 1, s = s + 1],

%t {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];

%t m = Map[t[#] &, Range[0, 40]]   (* A212894 *)

%t m/2 (* integers *)

%t 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 (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 Cf. A211795.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, May 30 2012