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A212561 Number of (w,x,y,z) with all terms in {1,...,n} and w + x = 2y + 2z.

%I A212561 #44 Aug 13 2025 10:21:31
%S A212561 0,0,1,5,12,26,45,75,112,164,225,305,396,510,637,791,960,1160,1377,
%T A212561 1629,1900,2210,2541,2915,3312,3756,4225,4745,5292,5894,6525,7215,
%U A212561 7936,8720,9537,10421,11340,12330,13357,14459,15600,16820,18081,19425
%N A212561 Number of (w,x,y,z) with all terms in {1,...,n} and w + x = 2y + 2z.
%C A212561 Probably related to A199771 and A200252.
%C A212561 For a guide to related sequences, see A211795.
%C A212561 Except for the first term, a(n) is the number of undirected rook moves on an n X n chessboard, considered up to rotations but not reflections. - _Hilko Koning_, Aug 10 2025
%H A212561 Colin Barker, <a href="/A212561/b212561.txt">Table of n, a(n) for n = 0..1000</a>
%H A212561 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A212561 a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6).
%F A212561 a(n) = (2*n^3-2*n^2+n-1-(n-1)*(-1)^n)/8 = (n-1)*(2*n^2+1-(-1)^n)/8. - _Luce ETIENNE_, Jul 26 2014
%F A212561 G.f.: x^2*(x^3+x^2+3*x+1) / ((x-1)^4*(x+1)^2). - _Colin Barker_, Feb 17 2015
%t A212561 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212561 (Do[If[w + x == 2 y + 2 z, s = s + 1],
%t A212561 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212561 Map[t[#] &, Range[0, 40]]   (* A212561 *)
%t A212561 LinearRecurrence[{2,1,-4,1,2,-1},{0,0,1,5,12,26},50] (* _Harvey P. Dale_, Dec 04 2016 *)
%t A212561 a[n_Integer?NonNegative] := ((n - 1) (2 n^2 + 1 - (-1)^n))/8
%t A212561 Table[a[n], {n, 0, 100}] (* _Hilko Koning_, Aug 10 2025 *)
%o A212561 (PARI) concat([0,0], Vec(x^2*(x^3+x^2+3*x+1)/((x-1)^4*(x+1)^2) + O(x^100))) \\ _Colin Barker_, Feb 17 2015
%Y A212561 Cf. A211795.
%K A212561 nonn,easy
%O A212561 0,4
%A A212561 _Clark Kimberling_, May 21 2012