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A336751
Smallest side of integer-sided triangles whose sides a < b < c are in arithmetic progression.
7
2, 3, 3, 4, 4, 5, 4, 5, 6, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14
OFFSET
1,1
COMMENTS
The triples of sides (a,b,c) with a < b < c are in increasing order of perimeter = 3*b, and if perimeter coincide, then by increasing order of the smallest side. This sequence lists the a's.
Equivalently: smallest side of integer-sided triangles such that b = (a+c)/2 with a < c.
a >= 2 and each side a appears a-1 times but not consecutively.
For each a = 3*k, k>=1, there exists exactly one right triangle (3*k, 4*k, 5*k) whose sides a < b < c are in arithmetic progression.
This sequence is not increasing a(6) = 5 for triangle with perimeter = 18 and a(7) = 4 for triangle with perimeter = 21. The smallest side is not an increasing function of the perimeter of these triangles.
For the corresponding triples and miscellaneous properties and references, see A336750.
REFERENCES
V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-290 p. 121, André Desvigne.
FORMULA
a(n) = A336750(n, 1).
EXAMPLE
a = 2 for only the smallest triangle (2, 3, 4).
a = 3 for Pythagorean triple (3, 4, 5) and also for the second triangle (3, 5, 7).
MAPLE
for b from 3 to 30 do
for a from b-floor((b-1)/2) to b-1 do
c := 2*b - a;
print(a);
end do;
end do;
MATHEMATICA
Flatten[Array[Range[#-Floor[(#-1)/2], #-1] &, 20, 3]] (* Paolo Xausa, Feb 28 2024 *)
CROSSREFS
Cf. A336750 (triples), this sequence (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter).
Cf. A335894 (smallest side when triangles angles are in arithmetic progression).
Sequence in context: A056792 A292127 A227861 * A294991 A300118 A256544
KEYWORD
nonn
AUTHOR
Bernard Schott, Aug 15 2020
STATUS
approved