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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.
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
OFFSET
1,2
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.
LINKS
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
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
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
(Python)
a227347 = [0]
for n in range(2, 50): a227347.append(a227347[-1] + 5*n**2//6)
print(a227347) # Gennady Eremin, Mar 13 2022
CROSSREFS
Cf. A171965.
Sequence in context: A077126 A172113 A172112 * A068043 A145069 A293350
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 08 2013
STATUS
approved