A126587 a(n) is the number of integer lattice points inside the right triangle with legs 3n and 4n (and hypotenuse 5n).

3, 17, 43, 81, 131, 193, 267, 353, 451, 561, 683, 817, 963, 1121, 1291, 1473, 1667, 1873, 2091, 2321, 2563, 2817, 3083, 3361, 3651, 3953, 4267, 4593, 4931, 5281, 5643, 6017, 6403, 6801, 7211, 7633, 8067, 8513, 8971, 9441, 9923, 10417, 10923, 11441

Offset: 1

Author

Zak Seidov, Jan 05 2007

Keywords

• nonn
• easy

Comments

Row sums of triangle A193832. - Omar E. Pol, Aug 22 2011

Examples

			At n=1, three lattice points (1,1), (1,2) and (2,1) are inside the triangle with vertices at the points (0,0), (3n,0) and (0,4n); hence a(1)=3.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Zak Seidov Inside points
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000326, A186424, A193832.

Programs

Magma

[6*n^2 - 4*n + 1: n in [1..50] ]; // Vincenzo Librandi, May 23 2011

Mathematica

nip[a_,b_]:=Sum[Floor[b-b*i/a-10^-6],{i,a-1}] Table[nip[3k,4k],{k,100}]
Table[6*n^2-4*n+1, {n,1,50}] (* G. C. Greubel, Mar 06 2018 *)

PARI

a(n)=6*n^2-4*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = A186424(2*n-1).

By Pick's theorem, a(n) = 6*n^2 - 4*n + 1. - Nick Hobson, Mar 13 2007

O.g.f.: x*(3+8*x+x^2)/(1-x)^3 = -1 - 12/(-1+x)^3 - 11/(-1+x) - 22/(-1+x)^2. - R. J. Mathar, Dec 10 2007

E.g.f.: exp(x)*(1 + 2*x + 6*x^2) - 1. - Stefano Spezia, May 09 2021

a(n) = (A000326(2n-1) + A000326(2n))/2. - Charlie Marion, Apr 17 2024