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A118058 a(n) = 49n^2 - 28n - 20.

%I A118058 #23 Sep 08 2022 08:45:24
%S A118058 1,120,337,652,1065,1576,2185,2892,3697,4600,5601,6700,7897,9192,
%T A118058 10585,12076,13665,15352,17137,19020,21001,23080,25257,27532,29905,
%U A118058 32376,34945,37612,40377,43240,46201,49260,52417,55672,59025,62476,66025
%N A118058 a(n) = 49n^2 - 28n - 20.
%C A118058 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(2)=A001652(2) and z(2)=A001653(2)=29; cf. A000290,A118057,A118059-A118061.
%H A118058 Vincenzo Librandi, <a href="/A118058/b118058.txt">Table of n, a(n) for n = 1..1000</a>
%H A118058 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A118058 a(n)+(a(n)+1)+...+(a(n)+98n+34)=7(7n-2)(7n+5)(14n+3)/2; e.g., 337+338+...+518=77805=7*19*26*45/2.
%F A118058 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+117*x-20*x^2)/(1-x)^3. - _Colin Barker_, Jun 30 2012
%e A118058 a(3)=49*3^2-28*3-20=337, a(4)=49*4^2-28*4-20=652 and 337+338+...+518=519+...+651.
%t A118058 Table[49*n^2 - 28*n - 20, {n, 10}] (* _Vincenzo Librandi_, Jul 08 2012 *)
%o A118058 (Magma) [49*n^2-28*n-20: n in [1..50]]; // _Vincenzo Librandi_, Jul 08 2012
%o A118058 (PARI) a(n)=49*n^2-28*n-20 \\ _Charles R Greathouse IV_, Jun 17 2017
%K A118058 nonn,easy
%O A118058 1,2
%A A118058 _Charlie Marion_, Apr 26 2006
%E A118058 Corrected by _Franklin T. Adams-Watters_ and _T. D. Noe_, Oct 25 2006