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A363269 Positive squares (A000290) alternating with positive square pyramidal numbers (A000330).

%I A363269 #26 Jul 30 2023 09:56:24
%S A363269 1,1,4,5,9,14,16,30,25,55,36,91,49,140,64,204,81,285,100,385,121,506,
%T A363269 144,650,169,819,196,1015,225,1240,256,1496,289,1785,324,2109,361,
%U A363269 2470,400,2870,441,3311,484,3795,529,4324,576,4900,625,5525,676,6201,729
%N A363269 Positive squares (A000290) alternating with positive square pyramidal numbers (A000330).
%H A363269 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).
%F A363269 a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8).
%F A363269 G.f.: x*(1 + x + x^3 - x^4)/(-1 + x^2)^4.
%F A363269 E.g.f.: ((18*x + 6*x^2)*cosh(x) + (6 + 6*x + 6*x^2 + x^3)*sinh(x))/24. - _Stefano Spezia_, Jun 10 2023
%F A363269 48*a(n) =  (n+1) * (n^2 +(-1)^n*n^2 -4*(-1)^n*n +8*n -6*(-1)^n +6). - _R. J. Mathar_, Jun 22 2023
%t A363269 c[1] = 1; c[2] = 1;
%t A363269 c[n_] := If[OddQ[n], c[n - 2] + n, c[n - 2] + c[n - 1]]
%t A363269 Table[c[n], {n, 1, 120}]
%o A363269 (PARI) a(n) = if(n%2, (n+1)^2/4, n*(n+1)*(n+2)/24); \\ _Kevin Ryde_, Jun 10 2023
%Y A363269 Cf. A000290, A000330, A123596, A363267, A363268.
%Y A363269 Range of terms: A363284\{0}.
%K A363269 nonn,easy
%O A363269 1,3
%A A363269 _Clark Kimberling_, May 24 2023