|
| |
|
|
A118059
|
|
288*n^2 - 168*n - 119.
|
|
3
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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(3)=A001652(3)=119 and z(3)=A001653(3)=169; cf. A000290, A118057-A118058, A118060-A118061.
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index to sequences with 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+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.
|
|
|
EXAMPLE
|
a(3)=288*3^2-168*3-119=337, a(4)=288*4^2-168*4-119=3817 and 1969+1970+...+3036=3037+...+3816
|
|
|
MATHEMATICA
|
Table[288*n^2 - 168*n - 119, {n, 100}] (* Vincenzo Librandi, Jul 08 2012 *)
|
|
|
PROG
|
(MAGMA) [288*n^2 - 168*n - 119: n in [1..50]]; // Vincenzo Librandi, Jul 08 2012
|
|
|
CROSSREFS
|
Sequence in context: A111105 A137559 A185377 * A028500 A133251 A116338
Adjacent sequences: A118056 A118057 A118058 * A118060 A118061 A118062
|
|
|
KEYWORD
|
nonn,easy,less
|
|
|
AUTHOR
|
Charlie Marion, Apr 26 2006
|
|
|
EXTENSIONS
|
Corrected by T. D. Noe, Nov 13 2006
|
|
|
STATUS
|
approved
|
| |
|
|
