This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227347 #47 Aug 05 2025 10:55:18
%S A227347 0,3,10,23,43,73,113,166,233,316,416,536,676,839,1026,1239,1479,1749,
%T A227347 2049,2382,2749,3152,3592,4072,4592,5155,5762,6415,7115,7865,8665,
%U A227347 9518,10425,11388,12408,13488,14628,15831,17098,18431,19831,21301,22841,24454
%N 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.
%C A227347 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.
%H A227347 Clark Kimberling, <a href="/A227347/b227347.txt">Table of n, a(n) for n = 1..1000</a>
%H A227347 Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.html">Inequalities and Identities Involving Sums of Integer Functions</a>, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
%H A227347 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,0,2,-1).
%F A227347 a(n) = Sum_{x=1..n} floor((5/6)*x^2).
%F A227347 a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7).
%F A227347 G.f.: (3*x^2 + 4*x^3 + 3*x^4)/((-1 + x)^4*(1 + 2*x + 2*x^2 + x^3)).
%F A227347 a(n) = (24*floor(n/3)+9*(-1)^n-9+(-32+(30+20*n)*n)*n)/72. - _Bruno Berselli_, Jul 09 2013
%e A227347 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.
%t A227347 z = 100; r = 5/6; k = 2; a[n_] := Sum[Floor[r*x^k], {x, 1, n}];
%t A227347 t = Table[a[n], {n, 1, z}]
%o A227347 (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
%o A227347 (Python)
%o A227347 a227347 = [0]
%o A227347 for n in range(2, 50): a227347.append(a227347[-1] + 5*n**2//6)
%o A227347 print(a227347) # _Gennady Eremin_, Mar 13 2022
%o A227347 (Python)
%o A227347 def A227347(n): return (24*(n//3)-(18 if n&1 else 0)+n*(n*(20*n+30)-32))//72 # _Chai Wah Wu_, Aug 05 2025
%Y A227347 Cf. A171965.
%K A227347 nonn,easy
%O A227347 1,2
%A A227347 _Clark Kimberling_, Jul 08 2013