Search: id:A016064
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%I A016064
%S A016064 1,3,13,51,193,723,2701,10083,37633,140451,524173,1956243,7300801,
%T A016064 27246963,101687053,379501251,1416317953,5285770563,19726764301,
%U A016064 73621286643,274758382273,1025412242451,3826890587533,14282150107683,53301709843201,
               198924689265123
%N A016064 Shortest legs of Heronian triangles (sides are consecutive integers, 
               area is an integer).
%C A016064 Least side in (m,m+1,m+2) integer-sided triangle with integer area.
%C A016064 Also describes triangles whose sides are consecutive integers and in 
               which an inscribed circle has an integer radius - Harvey P. Dale 
               (hpd1(AT)is2.nyu.edu), Dec 28 2000
%C A016064 Equivalently, positive integers m such that (3/16)*m^4 + (3/4)*m^3 + 
               (3/8)*m^2 - (3/4)*m - 9/16 is a square (A000290), a direct result 
               of Heron's formula. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), 
               Sep 04 2005
%F A016064 a(n) = 3 + floor((2+sqrt(3))*a(n-1)), n>=3. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), 
               Sep 04 2005
%F A016064 a(n) = 4*a(n-1) - a(n-2) + 2.
%F A016064 G.f.: (1-2x+3x^3)/((1-x)(1-4x+x^2))=(1-2x+3x^2)/(1-5x+5x^2-x^3); a(n)=(2+sqrt(3))^n+(2-sqrt(3))^n-1; 
               a(n)=2*A001075(n)-1. - Paul Barry (pbarry(AT)wit.ie), Feb 17 2004
%F A016064 Equals A003500(n)-1. - T. D. Noe (noe(AT)sspectra.com), Jun 17 2004
%F A016064 Ralf Stephan's formula (shown in his program) is the simplest (if proved) 
               not involving a recurrence or a different sequence.
%o A016064 (PARI) for(a=1,10^9, b=a+1; c=a+2; s=(a+b+c)/2; if(issquare(s*(s-a)*(s-b)*(s-c)), 
               print1(a,","))) (Shepherd)
%o A016064 (PARI) a(n)=if(n<1,1,-1+ceil((2+sqrt(3))^(n))) (from R. Stephan)
%Y A016064 Corresponding areas are in A011945.
%Y A016064 Cf. A001353, A019973 (2+sqrt(3)), A102341, A103974, A103975.
%Y A016064 Sequence in context: A008827 A026529 A101052 this_sequence A163774 A014985 
               A015521
%Y A016064 Adjacent sequences: A016061 A016062 A016063 this_sequence A016065 A016066 
               A016067
%K A016064 nonn
%O A016064 0,2
%A A016064 Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A016064 More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 18 
               2007

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