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

1, 697, 1969, 3817, 6241, 9241, 12817, 16969, 21697, 27001, 32881, 39337, 46369, 53977, 62161, 70921, 80257, 90169, 100657, 111721, 123361, 135577, 148369, 161737, 175681, 190201, 205297, 220969, 237217, 254041, 271441, 289417, 307969

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(3)=A001652(3)=119 and z(3)=A001653(3)=169; cf. A000290, A118057-A118058, A118060-A118061.

Examples

			a(3)=288*3^2-168*3-119=337, a(4)=288*4^2-168*4-119=3817 and 1969+1970+...+3036=3037+...+3816
		

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

[288*n^2 - 168*n - 119: n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

Mathematica

Table[288*n^2 - 168*n - 119, {n, 100}] (* Vincenzo Librandi, Jul 08 2012 *)

PARI

a(n)=288*n^2-168*n-119 \\ 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+694*x-119*x^2)/(1-x)^3. - Colin Barker, Jul 01 2012

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.

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.

Extensions

Corrected by T. D. Noe, Nov 13 2006