A024868 - OEIS

%I #39 Aug 11 2023 12:13:44

%S 6,8,22,27,52,61,100,114,170,190,266,293,392,427,552,596,750,804,990,

%T 1055,1276,1353,1612,1702,2002,2106,2450,2569,2960,3095,3536,3688,

%U 4182,4352,4902,5091,5700,5909,6580,6810,7546,7798,8602,8877,9752,10051,11000,11324,12350

%N a(n) = 2*(n+1) + 3*n + ... + (k+1)*(n+2-k), where k = floor(n/2).

%H Vincenzo Librandi, <a href="/A024868/b024868.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 a(n) = Sum_{i=1..floor(n/2)} (i+1)*(n-i+2) = floor(n/2)*(-2*floor(n/2)^2 + 3*n*floor(n/2) + 9*n + 14)/6, n>1. - _Wesley Ivan Hurt_, Sep 20 2013

%F G.f.: x^2*(6 + 2*x - 4*x^2 - x^3 + x^4) / ( (1+x)^3*(x-1)^4 ). - _R. J. Mathar_, Sep 25 2013

%F a(n) = 6*A058187(n-2) +2*A058187(n-3) -4*A058187(n-4) -A058187(n-5) +A058187(n-6). - _R. J. Mathar_, Sep 25 2013

%F a(n) = ( 4*n^3 + 33*n^2 + 38*n - 27 )/48 + (-1)^n*(n+3)^2/16. - _R. J. Mathar_, Sep 25 2013

%F E.g.f.: (1/24)*( x*(2*x^2 + 24*x + 27)*cosh(x) + (2*x^3 + 21*x^2 + 48*x - 27)*sinh(x) ). - _G. C. Greubel_, Jul 13 2022

%p seq(sum((i+1)*(k-i+2), i=1..floor(k/2)), k=2..70); # _Wesley Ivan Hurt_, Sep 20 2013

%t Table[Floor[n/2] (-2Floor[n/2]^2 +3n*Floor[n/2] +9n +14)/6, {n, 2, 100}] (* _Wesley Ivan Hurt_, Sep 20 2013 *)

%t CoefficientList[Series[(6 +2x -4x^2 -x^3 +x^4)/((1+x)^3 (1-x)^4), {x, 0, 60}], x] (* _Vincenzo Librandi_, Sep 26 2013 *)

%t LinearRecurrence[{1,3,-3,-3,3,1,-1},{6,8,22,27,52,61,100},50] (* _Harvey P. Dale_, Aug 11 2023 *)

%o (Magma) [19*n/24-9/16+n^3/12+11*n^2/16+(-1)^n*(3*n/8 +9/16+n^2/16): n in [2..50]]; // _Vincenzo Librandi_, Sep 26 2013

%o (SageMath) [(1/48)*(4*n^3 +33*n^2 +38*n -27 +3*(-1)^n*(n+3)^2) for n in (2..60)] # _G. C. Greubel_, Jul 13 2022

%Y Cf. A023855, A023856, A023857, A024305, A024854.

%K nonn,easy

%O 2,1

%A _Clark Kimberling_