|
|
A242497
|
|
Sides of (Heronian) triangles where sides are consecutive integers and area is an integer.
|
|
1
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|