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 A118061 #28 Jun 02 2025 00:29:18
%S A118061 1,23661,66921,129781,212241,314301,435961,577221,738081,918541,
%T A118061 1118601,1338261,1577521,1836381,2114841,2412901,2730561,3067821,
%U A118061 3424681,3801141,4197201,4612861,5048121,5502981,5977441,6471501
%N A118061 9800*n^2-5740*n-4059.
%C A118061 In general, all sequences of equations which contain every positive integer in order exactly once (a pairwise equal summed, ordered partition of the positive integers) may be defined as follows: For all k, let x(k)=A001652(k) and z(k)=A001653(k). Then if we define a(n) to be (x(k)+z(k))n^2-(z(k)-1)n-x(k), the following equation is true: a(n)+(a(n)+1)+...+(a(n)+(x(k)+z(k))n+(2x(k)+z(k)-1)/2)=(a(n)+ (x(k)+z(k))n+(2x(k)+z(k)+1)/2)+...+(a(n)+2(x(k)+z(k))n+x(k)); a(n)+2(x(k)+z(k))n+x(k))=a(n+1)-1; e.g., in this sequence, x(5)=A001652(5)=4059 and z(5)=A001653(5)=5741; cf. A000290,A118057-A118060.
%H A118061 Vincenzo Librandi, <a href="/A118061/b118061.txt">Table of n, a(n) for n = 1..1000</a>
%H A118061 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A118061 a(n)+(a(n)+1)+...+(a(n)+9800n+6929)=(a(n)+9800n+6930)+...+(a(n)+19600n+4059); a(n)+19600n+4059=a(n+1)-1; a(n+1)-1=a(n)+19600n+4059.
%F A118061 a(n)+(a(n)+1)+...+(a(n)+576n+203)=35(140n-41)(140n+29)(140n+99); e.g., 66921+66922+...+103250=3091156215=35*379*449*519.
%F A118061 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+23658*x-4059*x^2)/(1-x)^3. - _Colin Barker_, Jul 01 2012
%e A118061 a(3)= 9800*3^2-5740*3-4059=66921, a(4)=9800*4^2-5740*4-4059=129781 and 66921+66922+...+103250=103251+...+129780
%t A118061 CoefficientList[Series[(1+23658*x-4059*x^2)/(1-x)^3,{x,0,40}],x] (* _Vincenzo Librandi_, Jul 09 2012 *)
%o A118061 (Magma) [9800*n^2-5740*n-4059: n in [1..40]]; // _Vincenzo Librandi_, Jul 09 2012
%o A118061 (PARI) a(n)=9800*n^2-5740*n-4059 \\ _Charles R Greathouse IV_, May 10 2016
%K A118061 nonn,easy,less
%O A118061 1,2
%A A118061 _Charlie Marion_, Apr 26 2006
%E A118061 Corrected by _T. D. Noe_, Nov 13 2006