A242497 Sides of (Heronian) triangles where sides are consecutive integers and area is an integer.

3, 4, 5, 13, 14, 15, 51, 52, 53, 193, 194, 195, 723, 724, 725, 2701, 2702, 2703, 10083, 10084, 10085, 37633, 37634, 37635, 140451, 140452, 140453, 524173, 524174, 524175, 1956243, 1956244, 1956245, 7300801, 7300802, 7300803, 27246963, 27246964, 27246965

Offset: 1

Author

R. K. Guy and Charles R Greathouse IV, May 16 2014

Keywords

• nonn
• easy

Comments

Let the edge lengths of the triangle be 2x-1, 2x, 2x+1 so that area = sqrt{3x * x * (x-1) * (x+1)} and we need x^2 - 1 to be of shape 3y^2. That is, x/y is an even rank convergent to the continued fraction of sqrt(3) and x is A001075.

The intermediate length sides are given by A003500(n), n >= 1. Note that A003500(0) = 2 corresponds to the degenerate (Heronian) triangle with sides {1, 2, 3} and area 0. - Daniel Forgues, May 28 2014

References

• Nakane Genkei (Nakane the Elder), Shichijo Beki Yenshiki, 1691.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• David Eugene Smith and Yoshio Mikami, A history of Japanese mathematics, Dover, 2004, p. 168.
• Index entries for linear recurrences with constant coefficients, signature (-1,-1,4,4,4,-1,-1,-1).

Crossrefs

A016064 is the main entry for this sequence.

Programs

Mathematica

LinearRecurrence[{-1,-1,4,4,4,-1,-1,-1},{3,4,5,13,14,15,51,52},40] (* Harvey P. Dale, May 04 2021 *)

PARI

Vec((-3*x^7 - 5*x^6 - 6*x^5 + 4*x^4 + 10*x^3 + 12*x^2 + 7*x + 3)/(x^8 + x^7+ x^6 - 4*x^5 - 4*x^4 - 4*x^3 + x^2 + x + 1)+O(x^99))

Formula

G.f.: (-3*x^7 - 5*x^6 - 6*x^5 + 4*x^4 + 10*x^3 + 12*x^2 + 7*x + 3)/ ((1+x+x^2)*(1-4*x^3+x^6)). - R. J. Mathar, May 30 2023