Nonagonal Triangular Number
المؤلف:
Sloane, N. J. A
المصدر:
Sequences A048907, A048908, and A048909 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
19-12-2020
1331
Nonagonal Triangular Number
A number which is simultaneously a nonagonal number 
and a triangular number 
and therefore satisfies the Diophantine equation.
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Defining 
and 
gives the Pell-like equation
 |
(3)
|
This has unit solutions 
, (9, 3), and (19, 7), which lead to the family of solutions (5, 1), (9, 3), (19, 7), (61, 23), (135, 51), (299, 113), (971, 367), .... The corresponding integer solutions in 
and 
are 
, (10, 25), (154, 406), (2449, 6478), ... (OEIS A048907 and A048908), giving the nonagonal triangular numbers 1, 325, 82621, 20985481, 5330229625, 1353857339341, ... (OEIS A048909).
REFERENCES:
Sloane, N. J. A. Sequences A048907, A048908, and A048909 in "The On-Line Encyclopedia of Integer Sequences."
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