A115006 Row 2 of array in A114999.

0, 3, 8, 16, 26, 39, 54, 72, 92, 115, 140, 168, 198, 231, 266, 304, 344, 387, 432, 480, 530, 583, 638, 696, 756, 819, 884, 952, 1022, 1095, 1170, 1248, 1328, 1411, 1496, 1584, 1674, 1767, 1862, 1960, 2060, 2163, 2268, 2376, 2486, 2599, 2714, 2832, 2952, 3075, 3200

Offset

0,2

Comments

Number of lattice points (x,y) in the region of the coordinate plane bounded by y < 3x+1, y > x/2 and x <= n. - Wesley Ivan Hurt, Oct 27 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

a(n) = floor((n+1)^2/4)+n*(n+1).

G.f.: x*(2*x+3)/((1-x)^3*(1+x)).

From Wesley Ivan Hurt, Oct 27 2014: (Start)

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).

a(n) = (10*n^2 + 12*n + 1 - (-1)^n)/8.

a(n) = Sum_{i=1..n+1} (10*i + (-1)^i - 9)/4. (End)

E.g.f.: (x*(11 + 5*x)*cosh(x) + (1 + 11*x + 5*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Maple

A115006:=n->(10*n^2 + 12*n + 1 - (-1)^n)/8: seq(A115006(n), n=0..50); # Wesley Ivan Hurt, Oct 27 2014

Mathematica

Table[(10*n^2 + 12*n + 1 - (-1)^n)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 27 2014 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 3, 8, 16}, 60] (* Harvey P. Dale, Jan 13 2015 *)

Prog

(Magma) [ n*(n+1) + (n+1)^2 div 4: n in [0..50] ];
(PARI) {for(n=0, 50, print1(n*(n+1)+floor((n+1)^2/4), ", "))}

Crossrefs

Cf. A114999, A000217 (triangular numbers), A002620 (quarter-squares), A001859 (triangular numbers plus quarter-squares), A017305 (10n+3), A147874 (zero followed by partial sums of A017305).

Partial Sums of A047218.

Sequence in context: A122794 A225268 A211481 * A211480 A122796 A104249

Adjacent sequences: A115003 A115004 A115005 * A115007 A115008 A115009

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 23 2006

Extensions

Edited by Klaus Brockhaus, Nov 18 2008

Status

approved