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 A099721 #66 Aug 08 2025 12:29:45
%S A099721 0,3,20,63,144,275,468,735,1088,1539,2100,2783,3600,4563,5684,6975,
%T A099721 8448,10115,11988,14079,16400,18963,21780,24863,28224,31875,35828,
%U A099721 40095,44688,49619,54900,60543,66560,72963,79764,86975,94608,102675,111188,120159,129600
%N A099721 a(n) = n^2*(2*n+1).
%C A099721 For a right triangle with sides of lengths 8*n^3 + 12*n^2 + 8*n + 2, 4*n^4 + 8*n^3 + 4*n^2, and 4*n^4 + 8*n^3 + 12*n^2 + 8*n + 2, dividing the area by the perimeter gives a(n). - _J. M. Bergot_, Jul 30 2013
%C A099721 This sequence is the difference between the centered icosahedral (or cuboctahedral) numbers (A005902(n)) and the centered octagonal pyramidal numbers (A000447(n+1)). - _Peter M. Chema_, Jan 09 2016
%C A099721 a(n) is the sum of the integers in the closed interval (n-1)*n to n*(n+1). - _J. M. Bergot_, Apr 19 2017
%H A099721 Vincenzo Librandi, <a href="/A099721/b099721.txt">Table of n, a(n) for n = 0..1000</a>
%H A099721 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A099721 G.f.: x*(3 + 8*x + x^2)/(x-1)^4.
%F A099721 a(n) = A024196(n) - A024196(n-1). - _Philippe Deléham_, May 07 2012
%F A099721 a(n) = ceiling(Sum_{i=n^2-(n-1)..n^2+(n-1)} s(i)), for n > 0 and integer i, where s(i) are the real solutions to x = i + sqrt(x), and the summation range excludes the integer solutions which occur where i is an oblong number (A002378). The fractional portion of the summation converges to 2/3 for large n. If s(i) is replaced with i, then the summation equals n^2*(2*n-1) = A015237. - _Richard R. Forberg_, Oct 15 2014
%F A099721 a(n) = A005902(n) - A000447(n+1). - _Peter M. Chema_, Jan 09 2016
%F A099721 From _Amiram Eldar_, May 17 2022: (Start)
%F A099721 Sum_{n>=1} 1/a(n) = Pi^2/6 + 4*log(2) - 4.
%F A099721 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - Pi - 2*log(2) + 4. (End)
%F A099721 From _Elmo R. Oliveira_, Aug 08 2025: (Start)
%F A099721 E.g.f.: x*(1 + 2*x)*(3 + x)*exp(x).
%F A099721 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A099721 a(n) = A000290(n)*A005408(n). (End)
%p A099721 A099721 := proc(n) n^2*(2*n+1) ; end proc:
%p A099721 seq(A099721(n),n=0..10) ;
%t A099721 a[n_]:=2*n^3+n^2; (* _Vladimir Joseph Stephan Orlovsky_, Dec 21 2008 *)
%t A099721 LinearRecurrence[{4,-6,4,-1},{0,3,20,63},40] (* _Harvey P. Dale_, Aug 19 2022 *)
%o A099721 (Magma) [n^2*(2*n+1): n in [0..50]]; // _Vincenzo Librandi_, May 01 2011
%o A099721 (PARI) a(n) = ceil(sum(i=n^2-(n-1), n^2+(n-1), if(!issquare(4*i+1), (2*i+1+sqrt(4*i+1))/2, 0))); \\ _Michel Marcus_, Nov 14 2014, after _Richard R. Forberg_
%Y A099721 Row n=3 of A229079.
%Y A099721 Cf. A000578, A001093, A011379, A015237, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568, A098547.
%Y A099721 Cf. A000290, A000447, A002378, A005408, A005902, A024196.
%K A099721 nonn,easy
%O A099721 0,2
%A A099721 Douglas Winston (douglas.winston(AT)srupc.com), Nov 07 2004