A319258 a(n) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + ... + (up to n).

1, 3, 7, 11, 16, 41, 48, 56, 120, 130, 141, 262, 275, 289, 485, 501, 518, 807, 826, 846, 1246, 1268, 1291, 1820, 1845, 1871, 2547, 2575, 2604, 3445, 3476, 3508, 4532, 4566, 4601, 5826, 5863, 5901, 7345, 7385, 7426, 9107, 9150, 9194, 11130, 11176, 11223

Offset: 1

Author

Wesley Ivan Hurt, Sep 16 2018

Keywords

• nonn
• easy

Examples

			a(1) = 1;
a(2) = 1 + 2 = 3;
a(3) = 1 + 2*3 = 7;
a(4) = 1 + 2*3 + 4 = 11;
a(5) = 1 + 2*3 + 4 + 5 = 16;
a(6) = 1 + 2*3 + 4 + 5*6 = 41;
a(7) = 1 + 2*3 + 4 + 5*6 + 7 = 48;
a(8) = 1 + 2*3 + 4 + 5*6 + 7 + 8 = 56;
a(9) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 = 120;
a(10) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 = 130;
a(11) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11 = 141;
a(12) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 = 262; 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. A093361, A228958, A305189, A319014.

Programs

Mathematica

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}]

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

Formula

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.

From Colin Barker, Sep 16 2018: (Start)

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).

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)