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Nonagonal Triangular Number
A number which is simultaneously anonagonal number
and atriangular number
and therefore satisfies theDiophantine equation.
 | (1) |
Completing the square and rearranging gives
 | (2) |
Defining
and
gives the Pell-like equation
 | (3) |
This has unit solutions
, (9, 3), and (19, 7), which lead to the family of solutions (5, 1), (9, 3), (19, 7), (61, 23), (135, 51), (299, 113), (971, 367), .... The corresponding integer solutions in
and
are
, (10, 25), (154, 406), (2449, 6478), ... (OEISA048907 andA048908), giving the nonagonal triangular numbers 1, 325, 82621, 20985481, 5330229625, 1353857339341, ... (OEISA048909).
See also
Nonagonal Number,TriangularNumberExplore with Wolfram|Alpha
More things to try:
References
Sloane, N. J. A. SequencesA048907,A048908, andA048909 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Nonagonal Triangular NumberCite this as:
Weisstein, Eric W. "Nonagonal Triangular Number."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/NonagonalTriangularNumber.html