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A024163 Number of integer-sided triangles with sides a,b,c, a such that c - b < b - a.

%I A024163 #23 Aug 21 2025 11:16:01
%S A024163 0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,1,3,1,3,3,3,3,6,3,6,6,6,6,10,6,10,10,
%T A024163 10,10,15,10,15,15,15,15,21,15,21,21,21,21,28,21,28,28,28,28,36,28,36,
%U A024163 36,36,36,45,36,45,45,45,45,55,45,55,55,55,55,66,55,66,66,66,66,78,66,78,78,78,78,91
%N A024163 Number of integer-sided triangles with sides a,b,c, a<b<c, a+b+c=n such that c - b < b - a.
%C A024163 Number of partitions of n - 11 into parts 2, 3, and 6. - _Stefano Spezia_, Aug 21 2025
%H A024163 Vincenzo Librandi, <a href="/A024163/b024163.txt">Table of n, a(n) for n = 1..1000</a>
%H A024163 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,-1,1,0,-1,-1,0,1).
%F A024163 G.f.: x^11/((1-x^2)*(1-x^3)*(1-x^6)). - _Tani Akinari_, Oct 27 2014
%e A024163 2,4,5 for n=11 is the smallest such triangle.
%t A024163 CoefficientList[Series[x^10/((1-x^2)(1-x^3)(1-x^6)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Oct 28 2014 *)
%o A024163 (PARI) concat(vector(11,i,0), Vec(1/(1-x^2)/(1-x^3)/(1-x^6)+O(x^99))) \\ _Charles R Greathouse IV_, Oct 28 2014
%o A024163 (Sage)
%o A024163 def A024163_list(prec):
%o A024163     P.<x> = PowerSeriesRing(QQ, prec)
%o A024163     return P( x^11/((1-x^2)*(1-x^3)*(1-x^6)) ).list()
%o A024163 a=A024163_list(100); a[1:] # _G. C. Greubel_, Jul 03 2021
%o A024163 (Magma)
%o A024163 R<x>:=PowerSeriesRing(Rationals(), 100);
%o A024163 [0,0,0,0,0,0,0,0,0,0] cat Coefficients(R!( x^11/((1-x^2)*(1-x^3)*(1-x^6)) )); // _G. C. Greubel_, Jul 03 2021
%Y A024163 Cf. A024165, A025796.
%K A024163 nonn,easy
%O A024163 1,17
%A A024163 _Clark Kimberling_