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A024165 Number of integer-sided triangles with sides a,b,c, a such that c - b > b - a.

%I A024165 #32 Sep 01 2025 11:21:18
%S A024165 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,2,1,1,2,2,1,4,2,2,4,4,2,6,4,4,6,
%T A024165 6,4,9,6,6,9,9,6,12,9,9,12,12,9,16,12,12,16,16,12,20,16,16,20,20,16,
%U A024165 25,20,20,25,25,20,30,25,25,30,30,25,36,30,30,36,36,30,42,36,36,42,42,36,49,42,42,49,49
%N A024165 Number of integer-sided triangles with sides a,b,c, a<b<c, a+b+c=n such that c - b > b - a.
%C A024165 Same as A025828 with zeros prepended. - _Joerg Arndt_, Nov 04 2014
%H A024165 G. C. Greubel, <a href="/A024165/b024165.txt">Table of n, a(n) for n = 1..999</a>
%H A024165 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,0,1,-1,0,-1,-1,0,0,1).
%F A024165 G.f.: x^13/((1-x^3)*(1-x^4)*(1-x^6)). - _Tani Akinari_, Nov 04 2014
%F A024165 From _Robert Israel_, Nov 04 2014: (Start)
%F A024165 a(n) = a(n-3) + a(n-4) + a(n-6) - a(n-7) - a(n-9) - a(n-10) + a(n-13) for n >= 14.
%F A024165 a(6*n) = (2*n^2 - 8*n + 7 + (-1)^n)/8, n >= 1.
%F A024165 a(6*n+1) = a(6*n+4) = a(6*n+5) = (2*n^2 - 1 + (-1)^n)/8.
%F A024165 a(6*n+2) = a(6*n+3) = (2*n^2 - 4*n + 1 - (-1)^n)/8. (End)
%F A024165 From _Hoang Xuan Thanh_, Aug 31 2025: (Start)
%F A024165 a(n) = floor((n^2 -5*n +40 -(n-13)*(3*(-1)^n +8*((n+2) mod 3)) -12*((n+5) mod 6))/144).
%F A024165 a(n) = (floor((n-1)/3) - floor(n/4))*(floor((n-1)/3) + floor(n/4) - floor(n/2)). (End)
%t A024165 LinearRecurrence[{0,0,1,1,0,1,-1,0,-1,-1,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,1},100] (* _Harvey P. Dale_, Sep 04 2017 *)
%o A024165 (Sage)
%o A024165 def A024165_list(prec):
%o A024165     P.<x> = PowerSeriesRing(QQ, prec)
%o A024165     return P( x^13/((1-x^3)*(1-x^4)*(1-x^6)) ).list()
%o A024165 a=A024165_list(100); a[1:] # _G. C. Greubel_, Jul 03 2021
%o A024165 (Magma)
%o A024165 R<x>:=PowerSeriesRing(Rationals(), 100);
%o A024165 [0,0,0,0,0,0,0,0,0,0,0,0] cat Coefficients(R!( x^13/((1-x^3)*(1-x^4)*(1-x^6)) )); // _G. C. Greubel_, Jul 03 2021
%o A024165 (PARI) a(n) = ((n-1)\3 - n\4)*((n-1)\3 + n\4 - n\2) \\ _Hoang Xuan Thanh_, Aug 31 2025
%Y A024165 Cf. A024163, A024164, A025828.
%K A024165 nonn,easy,changed
%O A024165 1,19
%A A024165 _Clark Kimberling_