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Hexagonal Square Number

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Let H_n denote the nth hexagonal number and S_m the mth square number, then a number which is both hexagonal and square satisfies the equation H_n=S_m, or

n(2n-1)=m^2.
(1)

Completing the square and rearranging gives

(4n-1)^2-8m^2=1.
(2)

Therefore, defining

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

gives the Pell equation

x^2-2y^2=1.
(5)

The first few solutions are (x,y)=(3,2), (17, 12), (99, 70), (577, 408), .... These give the solutions (n,m)=(1,1), (9/2, 6), (25, 35), (289/2, 204), ..., giving the integer solutions (1, 1), (25, 35), (841, 1189), (28561, 40391), ... (OEIS A008844 and A046176). The corresponding hexagonal square numbers are 1, 1225, 1413721, 1631432881, 1882672131025, ... (OEIS A046177).

Closed-form solutions are

m=((3+2sqrt(2))^(2k+1)-(3-2sqrt(2))^(2k+1))/(4sqrt(2))
(6)
n=1/4{1+1/2[(3-2sqrt(2))^(2k+1)+(3+2sqrt(2))^(2k+1)]},
(7)

giving the kth hexagonal square number as

HS_k=1/(32)[-2+(17-12sqrt(2))(3-2sqrt(2))^(4k)+(17+12sqrt(2))(3+2sqrt(2))^(4k)].
(8)

A recurrence relation for m is given by

a_k=3a_(k-1)+sqrt(8a_(k-1)^2+1)
(9)

with a_1=1, where m_k=a_(2k-1) (M. Carreira, pers. comm., Sept. 11, 2004).

See also

Figurate Number, Hexagonal Number, Square Number

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References

Sloane, N. J. A. Sequences A008844, A046176, and A046177 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Hexagonal Square Number

Cite this as:

Weisstein, Eric W. "Hexagonal Square Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HexagonalSquareNumber.html

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