login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7).

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 A256525

Adjacent sequences:  A227344 A227345 A227346 * A227348 A227349 A227350

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 08 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified April 25 00:46 EDT 2017. Contains 285346 sequences.