Octagonal Square Number
المؤلف:
Graham, R. L.; Knuth, D. E.; and Patashnik, O
المصدر:
Concrete Mathematics: A Foundation for Computer Science. Reading, MA: Addison-Wesley
الجزء والصفحة:
...
20-12-2020
1568
Octagonal Square Number
A number which is simultaneously octagonal and square. Let 
denote the 
th octagonal number and 
the 
th square number, then a number which is both octagonal and square satisfies the equation 
, or
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Therefore, defining
gives the Pell equation
 |
(5)
|
The first few solutions are 
, (7, 4), (26, 15), (97, 56), (362, 209), (1351, 780), .... These give the solutions 
, (8/3, 4), (9, 15), (98/3, 56), (121, 209), ..., of which the integer solutions are (1, 1), (9, 15), (121, 209), (1681, 2911), ... (OEIS A046184 and A028230), corresponding to the octagonal square numbers 1, 225, 43681, 8473921, 1643897025, ... (OEIS A036428).
REFERENCES:
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science. Reading, MA: Addison-Wesley, p. 329, 1990.
Konhauser, J. D. E.; Velleman, D.; and Wagon, S. Which Way Did the Bicycle Go? And Other Intriguing Mathematical Mysteries. Washington, DC: Math. Assoc. Amer., p. 104, 1996.
Sloane, N. J. A. Sequences A028230, A036428, and A046184 in "The On-Line Encyclopedia of Integer Sequences."
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