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

1, 120, 337, 652, 1065, 1576, 2185, 2892, 3697, 4600, 5601, 6700, 7897, 9192, 10585, 12076, 13665, 15352, 17137, 19020, 21001, 23080, 25257, 27532, 29905, 32376, 34945, 37612, 40377, 43240, 46201, 49260, 52417, 55672, 59025, 62476, 66025

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(2)=A001652(2) and z(2)=A001653(2)=29; 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)+(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.

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

Example

a(3)=49*3^2-28*3-20=337, a(4)=49*4^2-28*4-20=652 and 337+338+...+518=519+...+651.

Mathematica

Table[49*n^2 - 28*n - 20, {n, 10}] (* Vincenzo Librandi, Jul 08 2012 *)

Prog

(Magma) [49*n^2-28*n-20: n in [1..50]]; // Vincenzo Librandi, Jul 08 2012
(PARI) a(n)=49*n^2-28*n-20 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Sequence in context: A098114 A135805 A327912 * A269037 A144524 A052768

Adjacent sequences: A118055 A118056 A118057 * A118059 A118060 A118061

Keyword

nonn,easy

Author

Charlie Marion, Apr 26 2006

Extensions

Corrected by Franklin T. Adams-Watters and T. D. Noe, Oct 25 2006

Status

approved