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A024164 Number of integer-sided triangles with sides a,b,c, a such that c - b = b - a.

%I A024164 #20 Mar 02 2024 13:12:48
%S A024164 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,
%T A024164 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,
%U A024164 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
%N A024164 Number of integer-sided triangles with sides a,b,c, a<b<c, a+b+c=n such that c - b = b - a.
%C A024164 From _Bernard Schott_, Oct 10 2020: (Start)
%C A024164 Equivalently: number of integer-sided triangles whose sides a < b < c are in arithmetic progression with perimeter n.
%C A024164 Equivalently: number of integer-sided triangles such that b = (a+c)/2 with a < c and perimeter n.
%C A024164 All the perimeters are multiple of 3 because each perimeter = 3 * middle side b.
%C A024164 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).
%C A024164 For the corresponding primitive triples and miscellaneous properties and references, see A336750. (End)
%H A024164 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%H A024164 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,0,0,1,0,0,-1).
%F A024164 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
%F A024164 From _Wesley Ivan Hurt_, Nov 01 2020: (Start)
%F A024164 G.f.: x^9/((x^3 - 1)^2*(x^3 + 1)).
%F A024164 a(n) = a(n-3) + a(n-6) - a(n-9).
%F A024164 a(n) = (1 - ceiling(n/3) + floor(n/3)) * floor((n-3)/6). (End)
%F A024164 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
%e A024164 a(9) = 1 for the smallest such triangle (2, 3, 4).
%e A024164 a(12) = 1 for Pythagorean triple (3, 4, 5).
%e A024164 a(15) = 2 for the two triples (3, 5, 7) and (4, 5, 6).
%t A024164 A024164[n_] := If[Mod[n, 3] == 0, Floor[(n - 3)/6], 0]; Array[A024164, 100] (* _Wesley Ivan Hurt_, Nov 01 2020 *)
%t A024164 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 *)
%Y A024164 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).
%Y A024164 Cf. A005044 (number of integer-sided triangles with perimeter = n).
%K A024164 nonn,easy
%O A024164 1,15
%A A024164 _Clark Kimberling_