A024862 - OEIS

%I #24 Apr 20 2023 02:27:55

%S 3,5,17,23,50,62,110,130,205,235,343,385,532,588,780,852,1095,1185,

%T 1485,1595,1958,2090,2522,2678,3185,3367,3955,4165,4840,5080,5848,

%U 6120,6987,7293,8265,8607,9690,10070,11270,11690,13013,13475,14927,15433,17020,17572,19300

%N a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = natural numbers, t = odd natural numbers.

%H Vincenzo Librandi, <a href="/A024862/b024862.txt">Table of n, a(n) for n = 2..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 G.f.: x^2*(3+2*x+3*x^2) / ((1+x)^3*(x-1)^4). - _R. J. Mathar_, Sep 25 2013

%F a(n) = 3*A058187(n-2) + 2*A058187(n-3) + 3*A058187(n-4). - _R. J. Mathar_, Sep 25 2013

%F From _Colin Barker_, Jan 29 2016: (Start)

%F a(n) = (8*n^3 + 6*(-1)^n*n^2 + 12*n^2 + 6*(-1)^n*n - 2*n + 3*(-1)^n - 3)/48.

%F a(n) = (4*n^3 + 9*n^2 + 2*n)/24 for n even.

%F a(n) = (4*n^3 + 3*n^2 - 4*n - 3)/24 for n odd. (End)

%F E.g.f.: (1/48)*(3*(1 - 4*x + 2*x^2)*exp(-x) + (-3 + 18*x + 36*x^2 + 8*x^3)*exp(x)). - _G. C. Greubel_, Apr 19 2023

%t CoefficientList[Series[(3+2x+3x^2)/((1+x)^3 (1-x)^4), {x,0,50}], x] (* _Vincenzo Librandi_, Sep 25 2013 *)

%o (PARI) Vec(x^2*(3+2*x+3*x^2)/((1+x)^3*(x-1)^4) + O(x^100)) \\ _Colin Barker_, Jan 29 2016

%o (Magma) [((2*n-1)*(2*n+1)*(2*n+3) +3*(-1)^n*(n^2+(n+1)^2))/48: n in [2..50]]; // _G. C. Greubel_, Apr 19 2023

%o (SageMath) [((2*n-1)*(2*n+1)*(2*n+3) +3*(-1)^n*(n^2+(n+1)^2))/48 for n in range(2,51)] # _G. C. Greubel_, Apr 19 2023

%Y Cf. A058187.

%K nonn,easy

%O 2,1

%A _Clark Kimberling_