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Partial sums of A230584.
1

%I #31 Jan 20 2018 09:11:19

%S 2,5,11,18,29,43,61,84,111,145,183,230,281,343,409,488,571,669,771,

%T 890,1013,1155,1301,1468,1639,1833,2031,2254,2481,2735,2993,3280,3571,

%U 3893,4219,4578,4941,5339,5741,6180,6623,7105,7591,8118,8649,9223,9801,10424

%N Partial sums of A230584.

%H Colin Barker, <a href="/A298375/b298375.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).

%F Let g = n + ((n + 1) mod 2), then for n > 1,

%F a(n) = (g^3 + 6*g^2 + 11*g + 18) / 12 - If(n mod 2 = 1, 0, ((n + 2) / 2)^2 + 2).

%F From _Colin Barker_, Jan 18 2018: (Start)

%F G.f.: x*(2 + x - x^2 - x^3 + 2*x^5 - x^6) / ((1 - x)^4*(1 + x)^2).

%F a(n) = (n^3 + 6*n^2 + 14*n) / 12 for n>1 and even.

%F a(n) = (n^3 + 6*n^2 + 11*n + 18) / 12 for n>1 and odd.

%F a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.

%F (End)

%e For n = 5 then a(5) = 2+3+6+7+11 = 29.

%t CoefficientList[ Series[(2 + x - x^2 - x^3 + 2x^5 - x^6)/((x -1)^4 (x + 1)^2), {x, 0, 50}], x] (* _Robert G. Wilson v_, Jan 18 2018 *)

%o (PARI) Vec(x*(2 + x - x^2 - x^3 + 2*x^5 - x^6) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ _Colin Barker_, Jan 18 2018

%Y Cf. A230584.

%K nonn,easy

%O 1,1

%A _Gerald Hillier_, Jan 18 2018