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

1,2

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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.

Example

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

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 *)

Prog

(Magma) [1681*n^2 - 984*n - 696: n in [1..40]]; // Vincenzo Librandi, Jul 09 2012
(PARI) a(n)=1681*n^2-984*n-696 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Sequence in context: A020430 A163010 A069332 * A146894 A013688 A221452

Adjacent sequences: A118057 A118058 A118059 * A118061 A118062 A118063

Keyword

nonn,easy,less

Author

Charlie Marion, Apr 26 2006

Extensions

Corrected by T. D. Noe, Nov 13 2006

Status

approved