Heptagonal Hexagonal Number
A number which is simultaneously a heptagonal number 👁 Hep_n
and hexagonal number 👁 Hex_m
. Such numbers exist when
Completing the square and rearranging gives
Substituting 👁 x=10n-3
and 👁 y=4m-1
gives the Pell-like quadratic Diophantine equation
| 👁 x^2-5y^2=4, |
(3)
|
which has solutions 👁 (x,y)=(3,1)
, (7, 3), (18, 8), (47, 21), (123, 55), .... The
integer solutions in 👁 m
and 👁 n
are then given by 👁 (n,m)=(1,1)
, (221, 247), (71065, 79453), (22882613, 25583539),
... (OEIS A048902 and A048901),
corresponding to the heptagonal hexagonal numbers 1, 121771, 12625478965, 1309034909945503,
... (OEIS A048903).
See also
Heptagonal Number, Hexagonal NumberExplore with Wolfram|Alpha
More things to try:
References
Sloane, N. J. A. Sequences A048901, A048902, and A048903 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Heptagonal Hexagonal NumberCite this as:
Weisstein, Eric W. "Heptagonal Hexagonal Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HeptagonalHexagonalNumber.html