A227347 Number of lattice points in the closed region bounded by the graphs of y = (5/6)*x^2, x = n, and y = 0, excluding points on the x-axis.

0, 3, 10, 23, 43, 73, 113, 166, 233, 316, 416, 536, 676, 839, 1026, 1239, 1479, 1749, 2049, 2382, 2749, 3152, 3592, 4072, 4592, 5155, 5762, 6415, 7115, 7865, 8665, 9518, 10425, 11388, 12408, 13488, 14628, 15831, 17098, 18431, 19831, 21301, 22841, 24454

Offset: 1

Author

Clark Kimberling, Jul 08 2013

Keywords

• nonn
• easy

Comments

Suppose that r is a rational number, k is a nonnegative integer, and let a(n) = Sum_{x = 1..n} floor(r*x^k). By the results in Mircea Merca's article, (a(n)) is linearly recurrent. Consequently, for integers b,c,u,v and polynomials p(x) <= q(x) with rational coefficients, the number a(n) of lattice points (x,y) in the closed (or open) region bounded by the vertical lines x = b*n + u, x = c*n + v and the graphs of y = p(x), y = q(x) gives a linearly recurrent sequence (a(n)). Likewise for regions bounded by two polynomial graphs, etc., as in A227347, A227353, and many other sequences.

Examples

			Example: Let R be the open region bounded by the graphs of y = (5/6)*x^2, x = n, and y = 0.  The line x = 1 has 0 = floor(5/6) lattice points in R; the line x = 2 has 3 = floor(20/6) lattice points; the line x = 3 has 10 = floor(20/6) + floor(45/6) lattice points.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Crossrefs

Cf. A171965.

Programs

Magma

[(2*n*(10*n^2+45*n+44)+24*Floor((n+1)/3)-9*(-1)^n+9)/72: n in [0..50]]; // Bruno Berselli, Jul 09 2013

Mathematica

z = 100; r = 5/6; k = 2; a[n_] := Sum[Floor[r*x^k], {x, 1, n}];
t = Table[a[n], {n, 1, z}]

Python

a227347 = [0]
for n in range(2, 50): a227347.append(a227347[-1] + 5*n**2//6)
print(a227347) # Gennady Eremin, Mar 13 2022

Python

def A227347(n): return (24*(n//3)-(18 if n&1 else 0)+n*(n*(20*n+30)-32))//72 # Chai Wah Wu, Aug 05 2025

Formula

a(n) = Sum_{x=1..n} floor((5/6)*x^2).

a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7).

G.f.: (3*x^2 + 4*x^3 + 3*x^4)/((-1 + x)^4*(1 + 2*x + 2*x^2 + x^3)).

a(n) = (24*floor(n/3)+9*(-1)^n-9+(-32+(30+20*n)*n)*n)/72. - Bruno Berselli, Jul 09 2013