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Number of (w,x,y,z) with all terms in {1,...,n} and 2|w-x|<n+|y-z|.
3

%I #10 Aug 01 2015 10:26:21

%S 0,1,12,67,212,527,1096,2045,3496,5621,8580,12591,17852,24627,33152,

%T 43737,56656,72265,90876,112891,138660,168631,203192,242837,287992,

%U 339197,396916,461735,534156,614811,704240,803121,912032,1031697

%N Number of (w,x,y,z) with all terms in {1,...,n} and 2|w-x|<n+|y-z|.

%C a(n)+A212688(n)=n^4.

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

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3, -1, -5, 5, 1, -3, 1).

%F a(n) = 3*a(n-1)-a(n-2)-5*a(n-3)+5*a(n-4)+a(n-5)-3*a(n-6)+a(n-7).

%F G.f.: (x + 9*x^2 + 32*x^3 + 28*x^4 + 13*x^5 + x^6)/(1 - 3*x + x^2 + 5*x^3 - 5*x^4 - x^5 + 3*x^6 - x^7).

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

%t (Do[If[2 Abs[w - x] < n + Abs[y - z], s = s + 1],

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

%t Map[t[#] &, Range[0, 40]] (* A212687 *)

%t LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {0, 1, 12, 67, 212, 527, 1096}, 40]

%Y Cf. A211795.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, May 25 2012