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A338896 Inradii of Pythagorean triples of A338895.

%I A338896 #46 Apr 17 2023 15:51:58
%S A338896 0,2,0,4,4,0,6,8,6,0,8,12,12,8,0,10,16,18,16,10,0,12,20,24,24,20,12,0,
%T A338896 14,24,30,32,30,24,14,0,16,28,36,40,40,36,28,16,0,18,32,42,48,50,48,
%U A338896 42,32,18,0,20,36,48,56,60,60,56,48,36,20,0
%N A338896 Inradii of Pythagorean triples of A338895.
%C A338896 Without the 0's, the sequence becomes 2*A003991. The 0 indices are triangular numbers A000217.
%H A338896 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%F A338896 When m and n define a row of triples in A338275 that gives rise to a triple (a row) of A338895, the current term corresponding to such a row is (m-n-1)*n/2.
%F A338896 If [a,b,c] is the n-th row of A338895, then a(n) = a*b/(a+b+c).
%F A338896 T(n, k) = 2*k*(n - k). Follows from the first comment. - _Peter Luschny_, Apr 17 2023
%e A338896 m 3
%e A338896 n 2  [0,4,4]    0
%e A338896 ------------------
%e A338896 m 5
%e A338896 n 2  [6,8,10]   2
%e A338896 n 4  [0,16,16]  0
%e A338896 ------------------
%e A338896 m 7
%e A338896 n 2  [16,12,20] 4
%e A338896 n 4  [10,24,26] 4
%e A338896 n 6  [0,36,36]  0
%e A338896 ------------------
%e A338896 m 9
%e A338896 n 2  [30,16,34] 6
%e A338896 n 4  [24,32,40] 8
%e A338896 n 6  [14,48,50] 6
%e A338896 n 8  [0,64,64]  0
%e A338896 ------------------.
%e A338896 The 7th row of A338895 is [30,16,34], so a(7) = 30*16/(30+16+34) = 6.
%e A338896 As a triangle:
%e A338896    0
%e A338896    2, 0
%e A338896    4, 4, 0
%e A338896    6, 8, 6, 0
%e A338896    8, 12, 12, 8, 0
%e A338896   10, 16, 18, 16, 10, 0
%e A338896   12, 20, 24, 24, 20, 12, 0,
%t A338896 Table[#1 #2/Total[{##}] & @@ {((#1 - 1)^2 - #2^2)/2, (#1 - 1) #2, ((#1 - 1)^2 + #2^2)/2} & @@ {m, n}, {m, 3, 23, 2}, {n, 2, m, 2}] // Flatten (* _Michael De Vlieger_, Dec 04 2020 *)
%o A338896 (PARI) lista(mm) = forstep (m=3, mm, 2, forstep (n=2, m, 2, my(v=[((m-1)^2 - n^2)/2, (m-1)*n, ((m-1)^2 + n^2)/2]); print1(v[1]*v[2]/vecsum(v), ", "))); \\ _Michel Marcus_, Dec 04 2020
%Y A338896 Cf. A338275, A338895.
%Y A338896 Cf. A003991 (see 2nd comment).
%K A338896 nonn,tabl
%O A338896 1,2
%A A338896 _David Lovler_, Nov 14 2020