A319258 - OEIS

%I #14 Sep 17 2018 08:32:32

%S 1,3,7,11,16,41,48,56,120,130,141,262,275,289,485,501,518,807,826,846,

%T 1246,1268,1291,1820,1845,1871,2547,2575,2604,3445,3476,3508,4532,

%U 4566,4601,5826,5863,5901,7345,7385,7426,9107,9150,9194,11130,11176,11223

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

%H Colin Barker, <a href="/A319258/b319258.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) = n*(1 + floor((n-2)/3) - floor(n/3)) + 3*floor(n/3)^2*(1 + floor(n/3)) + floor((n+2)/3)*(3*floor((n+2)/3) - 1)/2.

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

%F G.f.: x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((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 = 3;

%e a(3) = 1 + 2*3 = 7;

%e a(4) = 1 + 2*3 + 4 = 11;

%e a(5) = 1 + 2*3 + 4 + 5 = 16;

%e a(6) = 1 + 2*3 + 4 + 5*6 = 41;

%e a(7) = 1 + 2*3 + 4 + 5*6 + 7 = 48;

%e a(8) = 1 + 2*3 + 4 + 5*6 + 7 + 8 = 56;

%e a(9) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 = 120;

%e a(10) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 = 130;

%e a(11) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11 = 141;

%e a(12) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 = 262; etc.

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

%o (PARI) Vec(x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ _Colin Barker_, Sep 16 2018

%Y Cf. A093361, A228958, A305189, A319014.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Sep 16 2018