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%I #35 Sep 08 2022 08:46:02
%S 1,3,12,20,45,63,112,144,225,275,396,468,637,735,960,1088,1377,1539,
%T 1900,2100,2541,2783,3312,3600,4225,4563,5292,5684,6525,6975,7936,
%U 8448,9537,10115,11340,11988,13357,14079,15600,16400,18081,18963,20812,21780,23805
%N Number of (w,x,y,z) with all terms in {0,...,n}, w even, and x = y + z.
%C A signed version is A122576.
%C For a guide to related sequences, see A211795.
%C Partial sums of the positive elements of A129194. - _Omar E. Pol_, Dec 28 2013
%H Vincenzo Librandi, <a href="/A212760/b212760.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).
%F G.f.: ( 1+2*x+6*x^2+2*x^3+x^4 ) / ( (1+x)^3*(1-x)^4 ).
%F a(n) = (n+1)*(n+2)*(2*n+3+(-1)^n)/8. - _Wesley Ivan Hurt_, Jul 22 2014
%F a(n) = A260706(A001318(n+1). - _Reinhard Zumkeller_, Nov 17 2015
%F a(n) = Sum_{i=1..n+1} numerator(i^2/2). - _Wesley Ivan Hurt_, Feb 26 2017
%p A212760:=n->(n+1)*(n+2)*(2*n+3+(-1)^n)/8: seq(A212760(n), n=0..50); # _Wesley Ivan Hurt_, Jul 22 2014
%t t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[(Mod[w, 2] == 0) && x == y + z, s++],
%t {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t Map[t[#] &, Range[0, 50]] (* A212760 *)
%t Table[(n + 1) (n + 2) (2 n + 3 + (-1)^n)/8, {n, 0, 50}] (* _Wesley Ivan Hurt_, Jul 22 2014 *)
%t CoefficientList[Series[(1 + 2 x + 6 x^2 + 2 x^3 + x^4)/((1 + x)^3 (1 - x)^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 23 2014 *)
%o (Magma) [(n+1)*(n+2)*(2*n+3+(-1)^n)/8 : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 22 2014
%o (Haskell)
%o a212760 = a260706 . fromInteger . a001318 . (+ 1)
%o -- _Reinhard Zumkeller_, Nov 17 2015
%Y Cf. A211795.
%Y Cf. A001318, A260706.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, May 29 2012
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