%I #41 Sep 18 2018 06:26:45

%S 1,2,5,9,25,31,38,87,96,106,206,218,231,400,415,431,687,705,724,1085,

%T 1106,1128,1612,1636,1661,2286,2313,2341,3125,3155,3186,4147,4180,

%U 4214,5370,5406,5443,6812,6851,6891,8491,8533,8576,10425,10470,10516,12632

%N a(n) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 + ... + (up to n).

%H Colin Barker, <a href="/A305189/b305189.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,3,-3,0,-3,3,0,1,-1).

%F a(n) = 3*floor(n/3)*(floor(n/3) + 1)/2 + floor((n+1)/3)*(3*floor((n+1)/3)^2 - 1) + n*(floor((n-1)/3) - floor((n-2)/3)).

%F From _Colin Barker_, Sep 16 2018: (Start)

%F G.f.: x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3).

%F a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>10.

%F (End)

%e a(1) = 1;

%e a(2) = 1*2 = 2;

%e a(3) = 1*2 + 3 = 5;

%e a(4) = 1*2 + 3 + 4 = 9;

%e a(5) = 1*2 + 3 + 4*5 = 25;

%e a(6) = 1*2 + 3 + 4*5 + 6 = 31;

%e a(7) = 1*2 + 3 + 4*5 + 6 + 7 = 38;

%e a(8) = 1*2 + 3 + 4*5 + 6 + 7*8 = 87;

%e a(9) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 = 96;

%e a(10) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10 = 106;

%e a(11) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 = 206;

%e a(12) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 = 218; etc.

%p seq(coeff(series((x*(1+x+3*x^2+x^3+13*x^4-3*x^5-2*x^6+4*x^7))/((1-x)^4*(1+x+x^2)^3),x,n+1), x, n), n = 1 .. 50); # _Muniru A Asiru_, Sep 16 2018

%t Table[3*Floor[n/3]*(Floor[n/3] + 1)/2 + Floor[(n + 1)/3]*(3*Floor[(n + 1)/3]^2 - 1) + n*(Floor[(n - 1)/3] - Floor[(n - 2)/3]), {n, 50}]

%t LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1 }, {1, 2, 5, 9, 25, 31, 38, 87, 96, 106}, 50] (* _Stefano Spezia_, Sep 16 2018 *)

%o (PARI) Vec(x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ _Colin Barker_, Sep 16 2018

%Y Cf. A093361, A228958, A319014.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Sep 15 2018