A336900 a(n) is the first element of the lexicographically earliest Heron square root triple with area n > 0, or 0, if no such triple exists.

4, 5, 9, 10, 17, 15, 20, 20, 26, 25, 34, 30, 40, 35, 36, 40, 41, 45, 52, 50, 52, 55, 58, 60, 68, 65, 72, 65, 73, 72, 85, 80, 85, 82, 98, 89, 101, 95, 100, 100, 104, 104, 101, 110, 106, 115, 113, 117, 130, 125, 123, 128, 130, 135, 136, 130, 145, 145, 164, 144

Offset

1,1

Comments

A Heron square root triple is a triple of positive integers [a, b, c] with a >= b >= c such that Heron's formula applied to their square roots leads an integer value. This means: H(a, b, c) = sqrt(S*(S - A)*(S - B)*(S - C)) where A = sqrt(a), B = sqrt(b), C = sqrt(c) and S = (A + B + C)/2. We call H(a, b, c) the area of the triple [a, b, c].

Samuel Bodansky introduced Heron square root triple in A334818. The term 'Heron square root triple' is not standard.

Links

Table of n, a(n) for n=1..60.

Wikipedia, Heron's formula.

Wikipedia, Heronian triangle.

Example

a(1) = 4 because H[4, 2, 2] = sqrt((sqrt(2) + 1)*(sqrt(2) - 1)) = 1.
a(40) = 100 because H[100, 89, 89] = sqrt(25*(5 + sqrt(89))*(-5 + sqrt(89))) = sqrt(1600) = 40.
In both cases there are also no lexicographically earlier Heron square root triple that take this value.
.
The first few triples are:
[ 4,  2,  2] ->  1;   [40, 29, 25] -> 13;   [68, 58, 50] -> 25;
[ 5,  5,  4] ->  2;   [35, 35, 28] -> 14;   [65, 65, 52] -> 26;
[ 9,  8,  5] ->  3;   [36, 34, 34] -> 15;   [72, 65, 53] -> 27;
[10, 10,  8] ->  4;   [40, 40, 32] -> 16;   [65, 65, 64] -> 28;
[17, 13,  8] ->  5;   [41, 40, 37] -> 17;   [73, 68, 61] -> 29;
[15, 15, 12] ->  6;   [45, 45, 36] -> 18;   [72, 68, 68] -> 30;
[20, 17, 13] ->  7;   [52, 50, 34] -> 19;   [85, 68, 65] -> 31;
[20, 20, 16] ->  8;   [50, 50, 40] -> 20;   [80, 80, 64] -> 32;
[26, 20, 18] ->  9;   [52, 49, 45] -> 21;   [85, 73, 72] -> 33;
[25, 25, 20] -> 10;   [55, 55, 44] -> 22;   [82, 80, 74] -> 34;
[34, 26, 20] -> 11;   [58, 52, 50] -> 23;   [98, 82, 68] -> 35;
[30, 30, 24] -> 12;   [60, 60, 48] -> 24;   [89, 81, 80] -> 36;

Crossrefs

Cf. A334818.

Sequence in context: A329451 A308578 A034809 * A096808 A067261 A001161

Adjacent sequences: A336897 A336898 A336899 * A336901 A336902 A336903

Keyword

nonn

Author

Peter Luschny, Aug 07 2020

Status

approved