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
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