Octagonal Hexagonal Number
المؤلف:
Sloane, N. J. A.
المصدر:
Sequences A046190, A046191, and A046192 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
20-12-2020
1179
Octagonal Hexagonal Number
A number which is simultaneously octagonal and hexagonal. Let 
denote the 
th octagonal number and 
the 
th hexagonal number, then a number which is both octagonal and hexagonal satisfies the equation 
, or
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Therefore, defining
gives the second-order Diophantine equation
 |
(5)
|
The first few solutions are 
, (4, 3), (16, 13), (38, 31), (158, 129), (376, 307), .... These give the solutions 
, (1, 1), (3, 7/2), (20/3, 8), (80/3, 65/2), (63, 77), ..., of which the integer solutions are (1, 1), (63, 77), (6141, 7521), (601723, 736957), ... (OEIS A046190 and A046191), corresponding to the octagonal hexagonal numbers 1, 11781, 113123361, 1086210502741, ... (OEIS A046192).
REFERENCES:
Sloane, N. J. A. Sequences A046190, A046191, and A046192 in "The On-Line Encyclopedia of Integer Sequences."
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