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A105520 Sums of area and perimeter of Pythagorean triples, sorted in increasing order, including duplicates.

%I A105520 #41 Sep 06 2023 13:41:26
%S A105520 18,48,60,90,100,140,144,180,210,270,280,288,294,320,360,378,448,462,
%T A105520 480,594,600,648,660,720,728,756,858,900,900,924,980,1008,1008,1078,
%U A105520 1080,1120,1170,1210,1260,1344,1496,1530,1530,1568,1584,1584,1680,1700,1728
%N A105520 Sums of area and perimeter of Pythagorean triples, sorted in increasing order, including duplicates.
%H A105520 Giovanni Resta, <a href="/A105520/b105520.txt">Table of n, a(n) for n = 1..10711</a> (terms < 10^7)
%H A105520 Douglas Butler, <a href="https://www.tsm-resources.com/tsm/lists/trip.html">Pythagorean triples [including multiples] up to 2100</a>, Oundle School iCT Training Centre.
%H A105520 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html#anyint">Pythagorean triples</a>.
%e A105520 a(28) = 900 = (18+80+82) + (18*80/2) for 18*18 + 80*80 = 82*82.
%e A105520 a(29) = 900 = (25+60+65) + (25*60/2) for 25*25 + 60*60 = 65*65.
%e A105520 a(32) = 1008 = (24+70+74) + (24*70/2) for 24*24 + 70*70 = 74*74.
%e A105520 a(33) = 1008 = (36+48+60) + (36*48/2) for 36*36 + 48*48 = 60*60.
%t A105520 L = {}; mx = 1728; Do[ Do[ If[ IntegerQ[z = Sqrt[x^2 + y^2]], v = x y/2 + x + y + z; If[v <= mx, AppendTo[L, v], Break[]]], {y, x-1}], {x, 4, 4 + (2 mx^2)^(1/3)}]; Sort@ L (* _Giovanni Resta_, Mar 16 2020 *)
%o A105520 (Rexx)
%o A105520 T. = 0                        ;  S = ''
%o A105520 do C = 1 to 999               ;  H = C*C
%o A105520    do D = 1 to C              ;  I = D*D
%o A105520       do E = 1 to D           ;  J = E*E
%o A105520          if I + J < H   then  iterate E
%o A105520          if I + J = H   then  do
%o A105520             K = T.0 + 1       ;  T.0 = K
%o A105520             P = C + D + E     ;  A = ( D * E ) / 2
%o A105520             T.K = right( A + P, 6 )
%o A105520             T.K = T.K '=' A '+' P '(' E '+' D '+' C ')'
%o A105520          end
%o A105520          leave E
%o A105520       end E
%o A105520    end D
%o A105520 end C
%o A105520 call KWIK 'T.' /* sort by A+P for area A and perimeter P */
%o A105520 Y = 0
%o A105520 do N = 1 to T.0 while length( S ) < 255
%o A105520    X = word( T.N, 1 )         ;  say T.N
%o A105520    if X <= Y   then  say 'dupe:' N - 1 N ':' Y X
%o A105520    S = S || ', ' || X         ;  Y = X
%o A105520 end N
%o A105520 say substr( S, 3 )            /* _Frank Ellermann_, Mar 02 2020 */
%Y A105520 Cf. A103605, A103606, A024364, A024406, A105521.
%K A105520 easy,nonn
%O A105520 1,1
%A A105520 _Alexandre Wajnberg_, May 02 2005
%E A105520 Corrected and extended by _Frank Ellermann_, Mar 02 2020