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

1,1

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).

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

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 *)

Prog

(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))

Crossrefs

A016064 is the main entry for this sequence.

Sequence in context: A289121 A060738 A090651 * A062201 A352908 A211518

Adjacent sequences: A242494 A242495 A242496 * A242498 A242499 A242500

Keyword

nonn,easy

Author

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

Status

approved