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

18, 48, 60, 90, 100, 140, 144, 180, 210, 270, 280, 288, 294, 320, 360, 378, 448, 462, 480, 594, 600, 648, 660, 720, 728, 756, 858, 900, 900, 924, 980, 1008, 1008, 1078, 1080, 1120, 1170, 1210, 1260, 1344, 1496, 1530, 1530, 1568, 1584, 1584, 1680, 1700, 1728

Offset

1,1

Links

Giovanni Resta, Table of n, a(n) for n = 1..10711 (terms < 10^7)

Douglas Butler, Pythagorean triples [including multiples] up to 2100, Oundle School iCT Training Centre.

Ron Knott, Pythagorean triples.

Example

a(28) = 900 = (18+80+82) + (18*80/2) for 18*18 + 80*80 = 82*82.
a(29) = 900 = (25+60+65) + (25*60/2) for 25*25 + 60*60 = 65*65.
a(32) = 1008 = (24+70+74) + (24*70/2) for 24*24 + 70*70 = 74*74.
a(33) = 1008 = (36+48+60) + (36*48/2) for 36*36 + 48*48 = 60*60.

Mathematica

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 *)

Prog

(Rexx)
T. = 0                        ;  S = ''
do C = 1 to 999               ;  H = C*C
   do D = 1 to C              ;  I = D*D
      do E = 1 to D           ;  J = E*E
         if I + J < H   then  iterate E
         if I + J = H   then  do
            K = T.0 + 1       ;  T.0 = K
            P = C + D + E     ;  A = ( D * E ) / 2
            T.K = right( A + P, 6 )
            T.K = T.K '=' A '+' P '(' E '+' D '+' C ')'
         end
         leave E
      end E
   end D
end C
call KWIK 'T.' /* sort by A+P for area A and perimeter P */
Y = 0
do N = 1 to T.0 while length( S ) < 255
   X = word( T.N, 1 )         ;  say T.N
   if X <= Y   then  say 'dupe:' N - 1 N ':' Y X
   S = S || ', ' || X         ;  Y = X
end N
say substr( S, 3 )            /* Frank Ellermann, Mar 02 2020 */

Crossrefs

Cf. A103605, A103606, A024364, A024406, A105521.

Sequence in context: A218630 A099119 A360589 * A067726 A234956 A135189

Adjacent sequences: A105517 A105518 A105519 * A105521 A105522 A105523

Keyword

easy,nonn

Author

Alexandre Wajnberg, May 02 2005

Extensions

Corrected and extended by Frank Ellermann, Mar 02 2020

Status

approved