%I A143936
%S A143936 5262,2262756,972979926,418379105532,179902042398942,77357459852439636
%N A143936 Subsequence of A050791, "Fermat near misses", generated by iteration 
               of a linear form derived from Ramanujan's parametric formula for 
               equal sums of two pairs of cubes.
%C A143936 The formulae give an approximately geometric progression of values, z, 
               such that 1 + z^3 = x^3 + y^3, along with the values for x and y. 
               Iteration yields large values of x,y and z presumably unobtainable 
               by exhaustive search.
%D A143936 The Early Mathematics of Leonhard Euler By Charles Edward Sandifer, pp. 
               102-103
%H A143936 <a href="http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html">
               Wolfram Mathworld, Diophantine Equation 3rd Powers</a>
%H A143936 <a href="http://books.google.com/books?id=f9lAHgWV0vIC&printsec=frontcover#PPA103,
               M1">The Early Mathematics of Leonhard Euler p. 103, on Google Books</
               a>
%F A143936 In Ramanujan's parametric formula:
%F A143936 (a*x+y)^3 + (b+x^2*y)^3 = (b*x+y)^3 + (a+x^2*y)^3
%F A143936 with
%F A143936 a^2 + a*b + b^2 = x*y^2,
%F A143936 we set x=3, ax+y=1 and obtain a quadratic equation for b in terms of 
               a
%F A143936 ( Since 'a' is always negative we write it explicitly as '-a' and solve 
               for positive 'a' )
%F A143936 The surd of the quadratic formula then becomes:
%F A143936 sqrt(321*a^2 + 216*a + 36)
%F A143936 and we require that this be an integer. After finding an initial value 
               of 'a' which satisfies this condition by inspection of the sequence 
               A050791, we use Euler's method to find the bilinear recursion: ( 
               with s_i == sqrt(321*a_i^2 + 216*a_i + 36) )
%F A143936 a_i+1 = 215*a_i + 12*s_i + 72
%F A143936 s_i+1 = 215*s_i + 3852*a_i + 1296
%F A143936 and these yield the values of x,y and z from Ramanujan's formula.
%e A143936 1 + 5262^3 = 4528^3 + 3753^3 = 145697644729
%e A143936 1 + 2262756^3 = 1947250^3 + 1613673^3 = 11585457155467377217
%e A143936 1 + 972979926^3 = 837313192^3 + 693875529^3 = 921110304262410135315034777
%o A143936 (Other) /*
%o A143936 File: form.bc
%o A143936 Usage: bc form.bc
%o A143936 ( In UNIX shell, e.g. bash on Cygwin )
%o A143936 */
%o A143936 define a(x){ return( 321*x^2 + 216*x + 36 ); }
%o A143936 define b(x){ return( sqrt(a(x)) ); }
%o A143936 define n(z){ auto a,x; x=3; a = 215*z+12*b(z)+72 ;
%o A143936 a;b(a); return(v(a)); }
%o A143936 define v(z){ auto a,b,x,y,i,j,k,l;
%o A143936 a = z; b = ( a + b(a) )/2;
%o A143936 a = -a; x=3; y = 1-a*x;
%o A143936 i=a*x+y; j=b+x^2*y; k=b*x+y; l=a+x^2*y;
%o A143936 -a; b; i;j;k;l; i^3+j^3; k^3+l^3;
%o A143936 return ( -a ); }
%o A143936 z=144; v(z) ; z=n(z); z=n(z); z=n(z); /* ... etc. */
%Y A143936 A050791, A141326
%Y A143936 Sequence in context: A067224 A093182 A124658 this_sequence A053397 A133152 
               A031661
%Y A143936 Adjacent sequences: A143933 A143934 A143935 this_sequence A143937 A143938 
               A143939
%K A143936 nonn
%O A143936 1,1
%A A143936 Lewis Mammel (l_mammel(AT)att.net), Sep 05 2008