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A016064
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Shortest legs of Heronian triangles (sides are consecutive integers, area is an integer).
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10
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1, 3, 13, 51, 193, 723, 2701, 10083, 37633, 140451, 524173, 1956243, 7300801, 27246963, 101687053, 379501251, 1416317953, 5285770563, 19726764301, 73621286643, 274758382273, 1025412242451, 3826890587533, 14282150107683, 53301709843201, 198924689265123
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Least side in (m,m+1,m+2) integer-sided triangle with integer area.
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
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
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FORMULA
| a(n) = 3 + floor((2+sqrt(3))*a(n-1)), n>=3. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 04 2005
a(n) = 4*a(n-1) - a(n-2) + 2. a(n) = (2+sqrt(3))^n+(2-sqrt(3))^n-1. - Paul Barry (pbarry(AT)wit.ie), Feb 17 2004
For n>=1, a(n) = ceil((2+sqrt(3))^n)-1.
a(n) = 2*A001075(n)-1. - Paul Barry (pbarry(AT)wit.ie), Feb 17 2004
a(n) = A003500(n)-1. - T. D. Noe (noe(AT)sspectra.com), Jun 17 2004
G.f.: (1-2x+3x^3)/((1-x)(1-4x+x^2))=(1-2x+3x^2)/(1-5x+5x^2-x^3); - Paul Barry (pbarry(AT)wit.ie), Feb 17 2004
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PROG
| (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)
(PARI) a(n)=if(n<1, 1, -1+ceil((2+sqrt(3))^(n))) (from R. Stephan)
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CROSSREFS
| Corresponding areas are in A011945.
Cf. A001353, A019973 (2+sqrt(3)), A102341, A103974, A103975.
Sequence in context: A008827 A026529 A101052 * A163774 A197074 A014985
Adjacent sequences: A016061 A016062 A016063 * A016065 A016066 A016067
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KEYWORD
| nonn
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AUTHOR
| Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
| More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 18 2007
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