|
Search: id:A118057
|
|
|
| A118057 |
|
a(n)=8n^2-4n-3; a(n)+(a(n)+1)+...+(a(n)+8n+5)=(a(n)+8n+6)+...+a(n+1)-1; a(n+1)-1=a(n)+16n+3. |
|
+0
5
|
|
| 1, 21, 57, 109, 177, 261, 361, 477, 609, 757, 921, 1101, 1297, 1509, 1737, 1981, 2241, 2517, 2809, 3117, 3441, 3781, 4137, 4509, 4897, 5301, 5721, 6157, 6609, 7077, 7561, 8061, 8577, 9109, 9657, 10221, 10801, 11397, 12009, 12637, 13281, 13941, 14617
(list; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
In general, all sequences of equations which contain every positive integer in order exactly once (a pair wise 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(1)=A001652(1)=3 and z(1)=A001653(1)=5; cf. A000290,A118058-A118061.
|
|
FORMULA
|
a(n)+(a(n)+1)+...+(a(n)+8n+5)=(4n-1)(4n+1)(4n+3); e.g., 21+22+...+56=693=7*9*11.
|
|
EXAMPLE
|
a(3)=8*3^2-4*3-3=57, a(4)=8*4^2-4*4-3=109 and 57+58+...+86=87+...+108.
|
|
CROSSREFS
|
Sequence in context: A043382 A044123 A044504 this_sequence A020148 A037305 A051873
Adjacent sequences: A118054 A118055 A118056 this_sequence A118058 A118059 A118060
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Charlie Marion (charliemath(AT)optonline.net), Apr 26 2006
|
|
|
Search completed in 0.002 seconds
|
