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A212893 Number of quadruples (w,x,y,z) with all terms in {0,...,n} such that w-x, x-y, and y-z all have the same parity.

%I A212893 #31 Feb 21 2019 03:53:40
%S A212893 1,4,25,64,169,324,625,1024,1681,2500,3721,5184,7225,9604,12769,16384,
%T A212893 21025,26244,32761,40000,48841,58564,70225,82944,97969,114244,133225,
%U A212893 153664,177241,202500,231361,262144,297025,334084,375769
%N A212893 Number of quadruples (w,x,y,z) with all terms in {0,...,n} such that w-x, x-y, and y-z all have the same parity.
%C A212893 For a guide to related sequences, see A211795.
%C A212893 Sum of odd integers between 1 and (n+1)^2. - _Réjean Labrie_, Jan 14 2014
%H A212893 Muniru A Asiru, <a href="/A212893/b212893.txt">Table of n, a(n) for n = 0..10000</a>
%H A212893 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A212893 a(n) = (A000982(n+1))^2.
%F A212893 a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8).
%F A212893 G.f.: f(x)/g(x), where f(x) = -1 - 2*x - 15*x^2 - 12*x^3 - 15*x^4 - 2*x^5 - x^6 and g(x) = ((-1+x)^5)*(1+x)^3.
%p A212893 A212893 := n->ceil((n+1)^2/2)^2; seq(A212893(k), k=1..100); # _Wesley Ivan Hurt_, Jun 14 2013
%t A212893 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212893 (Do[If[Mod[w - x, 2] == Mod[x - y, 2] == Mod[y - z, 2], s = s + 1],
%t A212893 {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t A212893 m = Map[t[#] &, Range[0, 40]]   (* this sequence *)
%t A212893 Sqrt[m]  (* A000982 except for offset *)
%Y A212893 Cf. A000982, A211795.
%K A212893 nonn,easy
%O A212893 0,2
%A A212893 _Clark Kimberling_, May 30 2012