A213483 - OEIS

%I #17 Oct 27 2024 06:11:10

%S 1,5,13,23,38,55,78,103,135,169,211,255,308,363,428,495,573,653,745,

%T 839,946,1055,1178,1303,1443,1585,1743,1903,2080,2259,2456,2655,2873,

%U 3093,3333,3575,3838,4103,4390,4679,4991,5305,5643,5983,6348

%N Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| >= w+x+y.

%C a(n) + A213482(n) = (n+1)^3.

%C For a guide to related sequences, see A212959.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).

%F a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).

%F G.f.: (1 + 3*x + 2*x^2 - 4*x^3 - 2*x^4 + x^5)/((1 - x)^4*(1 + x)^2).

%F From _Ayoub Saber Rguez_, Dec 31 2021: (Start)

%F a(n) + A213482(n) = (n+1)^3.

%F a(n) = A213479(n) + A006918(n).

%F a(n)= (n^3 + 33*n^2 + 71*n + 15 + (3*n+9)*((n+1) mod 2))/24. (End)

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

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

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

%t Map[t[#] &, Range[0, 60]]   (* A213483 *)

%t LinearRecurrence[{2,1,-4,1,2,-1},{1,5,13,23,38,55},50] (* _Harvey P. Dale_, Sep 11 2019 *)

%Y Cf. A212959.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Jun 13 2012