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
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_{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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 08 2013
STATUS
approved