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A210375 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 3.

%I A210375 #20 Aug 08 2025 05:00:14
%S A210375 0,1,16,44,80,125,180,246,324,415,520,640,776,929,1100,1290,1500,1731,
%T A210375 1984,2260,2560,2885,3236,3614,4020,4455,4920,5416,5944,6505,7100,
%U A210375 7730,8396,9099,9840,10620,11440,12301,13204,14150,15140,16175,17256,18384,19560
%N A210375 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 3.
%C A210375 A210375 is also the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 3*n - 3.
%C A210375 See A210000 for a guide to related sequences.
%H A210375 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A210375 From _Colin Barker_, Dec 07 2017: (Start)
%F A210375 G.f.: x*(1 + 12*x - 14*x^2 - 4*x^3 + 6*x^4) / (1 - x)^4.
%F A210375 a(n) = (-120 + 74*n + 15*n^2 + n^3) / 6 for n > 1.
%F A210375 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 5. (End)
%t A210375 a = 0; b = n; z1 = 45;
%t A210375 t[n_] := t[n] = Flatten[Table[w + x + y + z, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A210375 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A210375 Table[c[n, n + 3], {n, 0, z1}]   (* A210375 *)
%t A210375 Table[c[n, 3 n - 3], {n, 0, z1}] (* A210375 *)
%Y A210375 Cf. A210000.
%K A210375 nonn,easy
%O A210375 0,3
%A A210375 _Clark Kimberling_, Mar 20 2012