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A210698 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 0 (mod 3).

%I A210698 #16 Mar 03 2024 18:40:18
%S A210698 1,8,33,90,209,528,889,1432,2673,3802,5297,8448,11025,14216,20625,
%T A210698 25546,31393,42768,51145,60824,79233,92394,107297,135168,154657,
%U A210698 176392,216513,244090,274481,330000,367641,408728,483153,533050,587089
%N A210698 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 0 (mod 3).
%C A210698 A210698(n)+2*A211071(n)=n^4.
%C A210698 For a guide to related sequences, see A210000.
%H A210698 Chai Wah Wu, <a href="/A210698/b210698.txt">Table of n, a(n) for n = 1..10000</a>
%H A210698 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1).
%F A210698 From _Chai Wah Wu_, Nov 30 2016: (Start)
%F A210698 a(n) = a(n-1) + 4*a(n-3) - 4*a(n-4) - 6*a(n-6) + 6*a(n-7) + 4*a(n-9) - 4*a(n-10) - a(n-12) + a(n-13) for n > 13.
%F A210698 G.f.: x*(-x^11 - 9*x^10 - 23*x^9 - 115*x^8 - 109*x^7 - 139*x^6 - 219*x^5 - 91*x^4 - 53*x^3 - 25*x^2 - 7*x - 1)/((x - 1)^5*(x^2 + x + 1)^4).
%F A210698 If r = floor(n/3), s = floor((n-1)/3)+1 and t = floor((n-2)/3)+1, then:
%F A210698 a(n) = r^4 + 4*r^3*s + 4*r^3*t + 4*r^2*s^2 + 8*r^2*s*t + 4*r^2*t^2 + s^4 + 6*s^2*t^2 + t^4.
%F A210698 If n == 0 mod 3, then a(n) = 11*n^4/27.
%F A210698 If n == 1 mod 3, then a(n) = (11*n^4 - 8*n^3 + 6*n^2 + 4*n + 14)/27.
%F A210698 If n == 2 mod 3, then a(n) = (11*n^4 - 16*n^3 + 24*n^2 + 32*n + 8)/27. (End)
%t A210698 a = 1; b = n; z1 = 45;
%t A210698 t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A210698 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A210698 u[n_] := u[n] = Sum[c[n, 3 k], {k, -2*n^2, 2*n^2}]
%t A210698 v[n_] := v[n] = Sum[c[n, 3 k + 1], {k, -2*n^2, 2*n^2}]
%t A210698 w[n_] := w[n] = Sum[c[n, 3 k + 2], {k, -2*n^2, 2*n^2}]
%t A210698 Table[u[n], {n, 1, z1}] (* A210698 *)
%t A210698 Table[v[n], {n, 1, z1}] (* A211071 *)
%t A210698 Table[w[n], {n, 1, z1}] (* A211071 *)
%t A210698 LinearRecurrence[{1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1}, {1, 8, 33, 90, 209, 528, 889, 1432, 2673, 3802, 5297, 8448, 11025}, 40] (* _Vincenzo Librandi_, Dec 01 2016 *)
%o A210698 (Python)
%o A210698 from __future__ import division
%o A210698 def A210698(n):
%o A210698     if n % 3 == 0:
%o A210698         return 11*n**4//27
%o A210698     elif n % 3 == 1:
%o A210698         return (11*n**4 - 8*n**3 + 6*n**2 + 4*n + 14)//27
%o A210698     else:
%o A210698         return (11*n**4 - 16*n**3 + 24*n**2 + 32*n + 8)//27 # _Chai Wah Wu_, Nov 30 2016
%Y A210698 Cf. A210000, A211071, A211033.
%K A210698 nonn
%O A210698 1,2
%A A210698 _Clark Kimberling_, Apr 01 2012