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Octagonal Triangular Number


A number which is simultaneously octagonal and triangular. Let O_n denote the nth octagonal number and T_m the mth triangular number, then a number which is both octagonal and triangular satisfies the equation O_n=T_m, or

 n(3n-2)=1/2m(m+1).
(1)

Completing the square and rearranging gives

 8(3n-1)^2-3(2m+1)^2=5.
(2)

Therefore, defining

x=2(2n-1)
(3)
y=2m+1
(4)

gives the second-order Diophantine equation

 2x^2-3y^2=5
(5)

The first few solutions are (x,y)=(2,1), (4, 3), (16, 13), (38, 31), (158, 129), (376, 307), .... These give the solutions (n,m)=(2/3,0), (1, 1), (3, 6), (20/3, 15), (80/3, 64), (63, 153), ..., of which the integer solutions are (1, 1), (3, 6), (63, 153), (261, 638), (6141, 15041), (25543, 62566), (601723, 1473913), ... (OEIS A046181 and A046182), corresponding to the octagonal triangular numbers 1, 21, 11781, 203841, 113123361, ... (OEIS A046183).


See also

Hexagonal Number, Octagonal Hexagonal Number, Pentagonal Number

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References

Sloane, N. J. A. Sequences A046181, A046182, and A046183 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Octagonal Triangular Number

Cite this as:

Weisstein, Eric W. "Octagonal Triangular Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OctagonalTriangularNumber.html

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