A118060 a(n) = 1681*n^2 - 984*n - 696.

1, 4060, 11481, 22264, 36409, 53916, 74785, 99016, 126609, 157564, 191881, 229560, 270601, 315004, 362769, 413896, 468385, 526236, 587449, 652024, 719961, 791260, 865921, 943944, 1025329, 1110076, 1198185, 1289656, 1384489, 1482684, 1584241

Offset: 1

Author

Charlie Marion, Apr 26 2006

Keywords

• nonn
• easy
• less

Comments

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.

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Magma

[1681*n^2 - 984*n - 696: n in [1..40]]; // Vincenzo Librandi, Jul 09 2012

Mathematica

CoefficientList[Series[(1+4057*x-696*x^2)/(1-x)^3,{x,0,40}],x] (* Vincenzo Librandi, Jul 09 2012 *)
LinearRecurrence[{3,-3,1},{1,4060,11481},40] (* Harvey P. Dale, Oct 28 2016 *)

PARI

a(n)=1681*n^2-984*n-696 \\ Charles R Greathouse IV, Jun 17 2017

Formula

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

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.

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.

Extensions

Corrected by T. D. Noe, Nov 13 2006