A118060 - OEIS

%I #28 Sep 08 2022 08:45:24

%S 1,4060,11481,22264,36409,53916,74785,99016,126609,157564,191881,

%T 229560,270601,315004,362769,413896,468385,526236,587449,652024,

%U 719961,791260,865921,943944,1025329,1110076,1198185,1289656,1384489,1482684,1584241

%N a(n) = 1681*n^2 - 984*n - 696.

%C 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(4)=A001652(4)=696 and z(4)=A001653(4)=985; cf. A000290, A118057-A118059, A118061.

%H Vincenzo Librandi, <a href="/A118060/b118060.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+4057*x-696*x^2)/(1-x)^3. - _Colin Barker_, Jul 01 2012

%F a(n)+(a(n)+1)+...+(a(n)+1681n+1188) = (a(n)+1681n+1189)+ ... +a(n+1)-1; a(n+1)-1 = a(n)+3362n+696.

%F a(n)+(a(n)+1)+...+(a(n)+1681n+1188)=41(41n-12)(41n+29)(82n+17)/2; e.g., 11481+11482+...+17712=90965388=41*111*152*263/2.

%e a(3)=1681*3^2-984*3-696=11481, a(4)=1681*4^2-984*4-696=22264 and 11481+11482+...+17712=17713+...+22263

%t CoefficientList[Series[(1+4057*x-696*x^2)/(1-x)^3,{x,0,40}],x] (* _Vincenzo Librandi_, Jul 09 2012 *)

%t LinearRecurrence[{3,-3,1},{1,4060,11481},40] (* _Harvey P. Dale_, Oct 28 2016 *)

%o (Magma) [1681*n^2 - 984*n - 696: n in [1..40]]; // _Vincenzo Librandi_, Jul 09 2012

%o (PARI) a(n)=1681*n^2-984*n-696 \\ _Charles R Greathouse IV_, Jun 17 2017

%K nonn,easy,less

%O 1,2

%A _Charlie Marion_, Apr 26 2006

%E Corrected by _T. D. Noe_, Nov 13 2006