cp's OEIS Frontend

A212967 Number of (w,x,y) with all terms in {0,...,n} and w < range{w,x,y}.

%I A212967 #33 Sep 08 2022 08:46:02
%S A212967 0,3,10,26,50,89,140,212,300,415,550,718,910,1141,1400,1704,2040,2427,
%T A212967 2850,3330,3850,4433,5060,5756,6500,7319,8190,9142,10150,11245,12400,
%U A212967 13648,14960,16371,17850,19434,21090,22857,24700,26660,28700
%N A212967 Number of (w,x,y) with all terms in {0,...,n} and w < range{w,x,y}.
%C A212967 For a guide to related sequences, see A212959.
%H A212967 Muniru A Asiru, <a href="/A212967/b212967.txt">Table of n, a(n) for n = 0..10000</a>
%H A212967 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A212967 a(n) + A212968(n) = (n + 1)^3.
%F A212967 a(n) = (n + 1)*(10*n*(n + 2) - 3*(-1)^n + 3)/24.
%F A212967 a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
%F A212967 G.f.: f(x)/g(x), where f(x) = x*(3 + 4*x + 3*x^2) and g(x) = ((1 - x)^4)(1 + x)^2.
%F A212967 a(n) = Sum_{k=1..n} A036666(k). - _Jon Maiga_, Nov 28 2018
%F A212967 E.g.f.: (exp(x)*(3 + 63*x + 60*x^2 + 10*x^3) - 3*exp(-x)*(1 - x))/24. - _Franck Maminirina Ramaharo_, Nov 29 2018
%p A212967 A212967:=n->(n+1)*(10*n*(n+2)-3*(-1)^n+3)/24: seq(A212967(n), n=0..100); # _Wesley Ivan Hurt_, Apr 28 2017
%t A212967 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212967 (Do[If[w < (Max[w, x, y] - Min[w, x, y]), s = s + 1],
%t A212967 {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
%t A212967 Map[t[#] &, Range[0, 60]]   (* A212967 *)
%t A212967 Accumulate[Accumulate[Table[n + LCM[n, 2], {n, 0, 60}]]] (* _Jon Maiga_, Nov 28 2018 *)
%t A212967 LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 3, 10, 26, 50, 89}, 50] (* _Vincenzo Librandi_, Nov 29 2018 *)
%o A212967 (GAP) List([1..45],n->Sum([1..n],k->(10*k*(k-1)+(2*k-1)*(-1)^k+1)/8)); # _Muniru A Asiru_, Nov 28 2018
%o A212967 (Magma) [(n+1)*(10*n*(n+2) - 3*(-1)^n+3)/24: n in [0..50]]; // _Vincenzo Librandi_, Nov 29 2018
%Y A212967 Cf. A212959, A212968, A036666.
%K A212967 nonn,easy
%O A212967 0,2
%A A212967 _Clark Kimberling_, Jun 02 2012