Nonagonal Hexagonal Number
المؤلف:
Sloane, N. J. A.
المصدر:
Sequences A048916, A048917, and A048918 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
18-12-2020
1118
Nonagonal Hexagonal Number
A number which is simultaneously a nonagonal number 
and hexagonal 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 fundamental solutions 
, (9, 3), and (19, 17), giving the family of solutions (5, 1), (9, 3), (19, 17), (61, 23), (135, 51), (509, 193), .... These give solutions which are integers in 
and 
of 
, (10, 13), (39025, 51625), ... (OEIS A048916 and A048917), giving the nonagonal hexagonal numbers 1, 325, 5330229625, 1353857339341, 22184715227362706161, ... (OEIS A048918).
REFERENCES:
Sloane, N. J. A. Sequences A048916, A048917, and A048918 in "The On-Line Encyclopedia of Integer Sequences."
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