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

Author

N. J. A. Sloane, Feb 23 2006

Keywords

• nonn
• easy

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).

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.

Programs

Magma

[ n*(n+1) + (n+1)^2 div 4: n in [0..50] ];

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 *)

PARI

{for(n=0, 50, print1(n*(n+1)+floor((n+1)^2/4), ","))}

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

Extensions

Edited by Klaus Brockhaus, Nov 18 2008