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A307481 Numbers that can be expressed as x+2y+z such that x, y, z, x+y, y+z, and x+2y+z are all positive squares.

%I A307481 #67 May 08 2019 15:49:50
%S A307481 625,2500,5625,10000,15625,22500,28561,30625,40000,50625,62500,75625,
%T A307481 83521,90000,105625,114244,122500,140625,142129,160000,180625,202500,
%U A307481 225625,250000,257049,275625,302500,330625,334084,360000,390625,422500,455625,456976,490000,525625
%N A307481 Numbers that can be expressed as x+2y+z such that x, y, z, x+y, y+z, and x+2y+z are all positive squares.
%C A307481 Generated by iterating through all combinations of x,y,z in the range 1..5000 (squared) and completing the addition pyramid (see Example section).
%C A307481 If k is in the sequence then so is k*m^2 for m >= 1. - _David A. Corneth_, May 04 2019
%C A307481 If a^2 + b^2 = c^2 then x = a^4, y = (ab)^2, z = b^4 gives a term x + 2y + z = c^4. - _David A. Corneth_, May 07 2019
%H A307481 David A. Corneth, <a href="/A307481/b307481.txt">Table of n, a(n) for n = 1..10575</a> (terms <= 3*10^10)
%H A307481 David A. Corneth, <a href="/A307481/a307481_2.png">Two examples in a pyramid shape</a>
%H A307481 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a307/A307481.java">Java program</a> (github)
%H A307481 Rémy Sigrist, <a href="/A307481/a307481.txt">C++ program for A307481</a>
%e A307481 Each addition pyramid is built up from three numbers x, y, and z as follows:
%e A307481 .
%e A307481        x+2y+z
%e A307481          / \
%e A307481         /   \
%e A307481       x+y   y+z
%e A307481       / \   / \
%e A307481      /   \ /   \
%e A307481     x     y     z
%e A307481 .
%e A307481 The first two terms, a(1)=625 and a(2)=2500, are the apex values for the first two pyramids consisting entirely of squares:
%e A307481 .
%e A307481          625                   2500
%e A307481          / \                    / \
%e A307481         /   \                  /   \
%e A307481       225   400              900  1600
%e A307481       / \   / \              / \   / \
%e A307481      /   \ /   \            /   \ /   \
%e A307481     81   144   256        324   576  1024
%o A307481 (Magma) a:=[]; for sw in [1..725] do w:=sw^2; for su in [1..Isqrt(w div 2)] do u:=su^2; v:=w-u; if IsSquare(v) then for sx in [1..Isqrt(u)] do x:=sx^2; y:=u-x; if (y gt 0) and IsSquare(y) then z:=v-y; if IsSquare(z) then a[#a+1]:=w; break su; end if; end if; end for; end if; end for; end for; a; // _Jon E. Schoenfield_, May 07 2019
%o A307481 (C++) See Links section.
%Y A307481 Cf. A000290 (squares).
%K A307481 nonn
%O A307481 1,1
%A A307481 _Glen Gilchrist_, Apr 10 2019