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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 30 07:41 EDT 2021. Contains 346348 sequences. (Running on oeis4.)