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A024164
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Number of integer-sided triangles with sides a,b,c, a<b<c, a+b+c=n such that c - b = b - a.
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6
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0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 5, 0, 0, 5, 0, 0, 6, 0, 0, 6, 0, 0, 7, 0, 0, 7, 0, 0, 8, 0, 0, 8, 0, 0, 9, 0, 0, 9, 0, 0, 10, 0, 0, 10, 0, 0, 11, 0, 0, 11, 0, 0, 12, 0, 0, 12, 0, 0, 13, 0, 0, 13, 0, 0, 14, 0, 0, 14, 0, 0, 15, 0, 0, 15, 0, 0, 16, 0, 0, 16
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OFFSET
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1,15
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COMMENTS
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Equivalently: number of integer-sided triangles whose sides a < b < c are in arithmetic progression with perimeter n.
Equivalently: number of integer-sided triangles such that b = (a+c)/2 with a < c and perimeter n.
All the perimeters are multiple of 3 because each perimeter = 3 * middle side b.
For each perimeter n = 12*k with k>0, there exists one and only one such right integer triangle whose triple is (3k, 4k, 5k).
For the corresponding primitive triples and miscellaneous properties and references, see A336750. (End)
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LINKS
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FORMULA
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If n = 3*k, then a(n) = floor((n-3)/6) = A004526((n-3)/3), otherwise, a(3k+1) = a(3k+2) = 0. - Bernard Schott, Oct 10 2020
G.f.: x^9/((x^3 - 1)^2*(x^3 + 1)).
a(n) = a(n-3) + a(n-6) - a(n-9).
a(n) = (1 - ceiling(n/3) + floor(n/3)) * floor((n-3)/6). (End)
E.g.f.: (18 + (x - 6)*cosh(x) + (x - 3)*sinh(x) - exp(-x/2)*((9 + 3*exp(x) + x)*cos(sqrt(3)*x/2) + sqrt(3)*x*sin(sqrt(3)*x/2)))/18. - Stefano Spezia, Feb 29 2024
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EXAMPLE
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a(9) = 1 for the smallest such triangle (2, 3, 4).
a(12) = 1 for Pythagorean triple (3, 4, 5).
a(15) = 2 for the two triples (3, 5, 7) and (4, 5, 6).
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MATHEMATICA
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LinearRecurrence[{0, 0, 1, 0, 0, 1, 0, 0, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, 120] (* Harvey P. Dale, Jun 03 2021 *)
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CROSSREFS
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Cf. A336750 (triples), A336751 (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter), this sequence (number of triangles whose perimeter = n), A336755 (primitive triples), A336756 (primitive perimeters), A336757 (number of primitive triangles with perimeter = n).
Cf. A005044 (number of integer-sided triangles with perimeter = n).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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