Nonagonal Square Number
المؤلف:
Sloane, N. J. A.
المصدر:
Sequences A048910, A048911, and A036411 in "The On-Line Encyclopedia of Integer Sequences.
الجزء والصفحة:
...
19-12-2020
1161
Nonagonal Square Number
A number which is simultaneously a nonagonal number 
and a square 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 
, (23, 6), and (75, 20), which lead to the family of solutions (9, 2), (23, 6), (75, 20), (247, 66), (681, 182), (2245, 600), .... The corresponding integer solutions in 
and 
are 
, (2, 3), (18, 33), (49, 91), (529, 989), ... (OEIS A048910 and A048911), giving the nonagonal square numbers 1, 9, 1089, 8281, 978121, 7436529, ... (OEIS A036411).
REFERENCES:
Sloane, N. J. A. Sequences A048910, A048911, and A036411 in "The On-Line Encyclopedia of Integer Sequences."
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