A212683 - OEIS

%I #21 Oct 06 2021 20:42:59

%S 0,0,2,8,22,46,84,138,212,308,430,580,762,978,1232,1526,1864,2248,

%T 2682,3168,3710,4310,4972,5698,6492,7356,8294,9308,10402,11578,12840,

%U 14190,15632,17168,18802,20536,22374,24318,26372,28538,30820

%N Number of (w,x,y,z) with all terms in {1,...,n} and |x-y| = w + |y-z|.

%C For a guide to related sequences, see A211795.

%C Also the number of (w,x,y) with all terms in {0,...,n-1} and |w-x| < |x-y|, see A212959. - _Clark Kimberling_, Jun 02 2012

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).

%F a(n) = 2*A019298(n-1) for n>=1.

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

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

%F a(n) + A212962(n-1) = n^3. - _Clark Kimberling, Jun 02 2012

%F a(n) = (2*n^3 - 3*n^2 + 2*n - (n mod 2))/4. - _Ayoub Saber Rguez_, Sep 02 2021

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

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

%t {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];

%t Map[t[#] &, Range[0, 40]]   (* A212683 *)

%t %/2  (* A019298 *)

%t LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 8, 22}, 40]

%Y Cf. A019298, A211795, A212962.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, May 24 2012