

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 xaxis.


2



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
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OFFSET

1,2


COMMENTS

Suppose that r is a rational number, k is a nonnegative integer, and let a(n) = Sum{floor(r*x^k), x = 1..n}. 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.


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).


FORMULA

a(n) = sum(floor((5/6)*x^2), x=1..n).
a(n) = 2*a(n1)  a(n3)  a(n4) + 2*a(n6)  a(n7).
G.f.: (3*x + 4*x^2 + 3*x^3)/((1 + x)^4*(1 + 2*x + 2*x^2 + x^3)).
a(n) = (2*n*(10*n^2+45*n+44) + 24*floor((n+1)/3)  9*(1)^n + 9)/72. [Bruno Berselli, Jul 09 2013]


EXAMPLE

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.


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}]


PROG

(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


CROSSREFS

Cf. A171965.
Sequence in context: A077126 A172113 A172112 * A068043 A145069 A293350
Adjacent sequences: A227344 A227345 A227346 * A227348 A227349 A227350


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Jul 08 2013


STATUS

approved



