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A118059 288*n^2 - 168*n - 119.

%I A118059 #24 Sep 08 2022 08:45:24
%S A118059 1,697,1969,3817,6241,9241,12817,16969,21697,27001,32881,39337,46369,
%T A118059 53977,62161,70921,80257,90169,100657,111721,123361,135577,148369,
%U A118059 161737,175681,190201,205297,220969,237217,254041,271441,289417,307969
%N A118059 288*n^2 - 168*n - 119.
%C A118059 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(3)=A001652(3)=119 and z(3)=A001653(3)=169; cf. A000290, A118057-A118058, A118060-A118061.
%H A118059 Vincenzo Librandi, <a href="/A118059/b118059.txt">Table of n, a(n) for n = 1..1000</a>
%H A118059 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A118059 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+694*x-119*x^2)/(1-x)^3. - _Colin Barker_, Jul 01 2012
%F A118059 a(n)+(a(n)+1)+...+(a(n)+288n+203)=(a(n)+288n+204)+...+a(n+1)-1; a(n+1)-1=a(n)+576n+119.
%F A118059 a(n)+(a(n)+1)+...+(a(n)+288n+203)=6(24n-7)(24n+5)(24n+17); e.g., 1969+1970+...+3036=2672670=6*65*77*89.
%e A118059 a(3)=288*3^2-168*3-119=337, a(4)=288*4^2-168*4-119=3817 and 1969+1970+...+3036=3037+...+3816
%t A118059 Table[288*n^2 - 168*n - 119, {n, 100}] (* _Vincenzo Librandi_, Jul 08 2012 *)
%o A118059 (Magma) [288*n^2 - 168*n - 119: n in [1..50]]; // _Vincenzo Librandi_, Jul 08 2012
%o A118059 (PARI) a(n)=288*n^2-168*n-119 \\ _Charles R Greathouse IV_, Jun 17 2017
%K A118059 nonn,easy,less
%O A118059 1,2
%A A118059 _Charlie Marion_, Apr 26 2006
%E A118059 Corrected by _T. D. Noe_, Nov 13 2006