A249547 a(n) = (10*n^2+8*n-1+(-1)^n)/8.

0, 2, 7, 14, 24, 36, 51, 68, 88, 110, 135, 162, 192, 224, 259, 296, 336, 378, 423, 470, 520, 572, 627, 684, 744, 806, 871, 938, 1008, 1080, 1155, 1232, 1312, 1394, 1479, 1566, 1656, 1748, 1843, 1940, 2040, 2142, 2247, 2354, 2464, 2576, 2691, 2808, 2928, 3050

Offset: 0

Author

Wesley Ivan Hurt, Oct 31 2014

Keywords

• nonn
• easy

Comments

a(n) is the number of lattice points (x,y) in the coordinate plane bounded by y < 3x, y >= x/2 and x <= n.

a(n)+1 is the number of lattice points bounded by y <= 3x, y >= x/2 and x <= n.

Links

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

Crossrefs

Cf. A001068, A004526, A135706, A168668, A226292.

Programs

Magma

[(10*n^2+8*n-1+(-1)^n)/8 : n in [0..50]];

Maple

A249547:=n->(10*n^2+8*n-1+(-1)^n)/8: seq(A249547(n), n=0..100);

Mathematica

Table[(10*n^2 + 8 n - 1 + (-1)^n)/8 , {n, 0, 50}]

PARI

a(n) = (10*n^2+8*n-1+(-1)^n)/8; \\ Michel Marcus, Nov 04 2014

PARI

concat(0, Vec(x*(2+3*x)/((1-x)^3*(1+x)) + O(x^100))) \\ Altug Alkan, Oct 28 2015

Formula

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

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

a(n) = A004526(n) + A226292(n), for n>0.

a(n) = Sum_{i=0..n} A001068(2*i). - Wesley Ivan Hurt, May 06 2016

E.g.f.: (x*(9 + 5*x)*exp(x) - sinh(x))/4. - Ilya Gutkovskiy, May 06 2016

a(2n) = A168668(n). a(2n-1) = A135706(n). - Wesley Ivan Hurt, May 09 2016