cp's OEIS Frontend

A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.

%I A227353 #35 Mar 24 2022 03:43:42
%S A227353 0,1,2,4,7,10,14,18,23,29,35,42,49,57,66,75,85,95,106,118,130,143,156,
%T A227353 170,185,200,216,232,249,267,285,304,323,343,364,385,407,429,452,476,
%U A227353 500,525,550,576,603,630,658,686,715,745,775,806,837,869,902,935
%N A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.
%C A227353 See A227347.
%H A227353 Clark Kimberling, <a href="/A227353/b227353.txt">Table of n, a(n) for n = 1..1000</a>
%H A227353 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,-2,1).
%F A227353 a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).
%F A227353 G.f.: (x*(1 + x^2 + x^3))/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
%F A227353 According to Wolfram Alpha, a(n) = floor(Re(E(n^2|Pi))) where E(x|m) is the incomplete elliptic integral of the second kind. - _Kritsada Moomuang_, Jan 28 2022
%F A227353 a(n) = a(n-1) + floor(3*n/5), n > 1. - _Gennady Eremin_, Mar 15 2022
%F A227353 a(n) = floor(n*(3*n-1)/10). - _Kevin Ryde_, Mar 15 2022
%e A227353 a(1) = floor(3/5) = 0; a(2) = floor(6/5) = 1; a(3) = a(2) + floor(9/5) = 2; a(4) = a(3) + floor(12/5) = 4.
%t A227353 z = 150; r = 3/5; k = 1; a[n_] := Sum[Floor[r*x^k], {x, 1, n}]; t = Table[a[n], {n, 1, z}]
%o A227353 (PARI) a(n) = (3*n^2-n)\10; \\ _Kevin Ryde_, Mar 15 2022
%o A227353 (Python) a = lambda n: n*(3*n-1)//10 # _Gennady Eremin_, Mar 20 2022
%Y A227353 Cf. A227347, A130520, A011858, A033437.
%Y A227353 Cf. A057355 (first differences).
%K A227353 nonn,easy
%O A227353 1,3
%A A227353 _Clark Kimberling_, Jul 08 2013