A319493 a(n) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10*11 - 12 + 13*14 - ... + (up to n).

1, 2, -1, 3, 19, 13, 20, 69, 60, 70, 170, 158, 171, 340, 325, 341, 597, 579, 598, 959, 938, 960, 1444, 1420, 1445, 2070, 2043, 2071, 2855, 2825, 2856, 3817, 3784, 3818, 4974, 4938, 4975, 6344, 6305, 6345, 7945, 7903, 7946, 9795, 9750, 9796, 11912, 11864, 11913

Offset: 1

Author

Wesley Ivan Hurt, Sep 20 2018

Keywords

• sign
• easy

Examples

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2 - 3 = -1;
a(4) = 1*2 - 3 + 4 = 3;
a(5) = 1*2 - 3 + 4*5 = 19;
a(6) = 1*2 - 3 + 4*5 - 6 = 13;
a(7) = 1*2 - 3 + 4*5 - 6 + 7 = 20;
a(8) = 1*2 - 3 + 4*5 - 6 + 7*8 = 69;
a(9) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 = 60;
a(10) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10 = 70;
a(11) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10*11 = 170;
a(12) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10*11 - 12 = 158;
a(13) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10*11 - 12 + 13 = 171;
a(14) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10*11 - 12 + 13*14 = 340; etc.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Crossrefs

Cf. A305189, A319258.

Programs

Mathematica

Table[Floor[(n + 1)/3]*(3*Floor[(n + 1)/3]^2 - 1) + n*(Floor[(n - 1)/3] - Floor[(n - 2)/3]) - 3*Floor[n/3]*(Floor[n/3] + 1)/2, {n, 50}]
CoefficientList[Series[(1 + x)*(1 - 3*x^2 + 4*x^3 + 9*x^4 - 6*x^5 + 4*x^6)/((1 - x)^4*(1 + x + x^2)^3), {x, 0, 50}], x] (* Stefano Spezia, Sep 23 2018 *)

PARI

Vec(x*(1 + x)*(1 - 3*x^2 + 4*x^3 + 9*x^4 - 6*x^5 + 4*x^6) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^50)) \\ Colin Barker, Sep 20 2018

Formula

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

From Colin Barker, Sep 20 2018: (Start)

G.f.: x*(1 + x)*(1 - 3*x^2 + 4*x^3 + 9*x^4 - 6*x^5 + 4*x^6) / ((1 - x)^4*(1 + x + x^2)^3).

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. (End)