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A151961
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Semiperimeter of the n-th Heronian triangle.
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3
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3, 6, 21, 78, 291, 1086, 4053, 15126, 56451, 210678, 786261, 2934366, 10951203, 40870446, 152530581, 569251878, 2124476931, 7928655846, 29590146453, 110431929966, 412137573411, 1538118363678, 5740335881301, 21423225161526, 79952564764803, 298387033897686
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OFFSET
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1,1
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COMMENTS
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The side lengths are consecutive integers (A016064) and the area is an integer (A011945).
Except for the first term, positive values of x (or y) satisfying x^2 - 4*x*y + y^2 + 27 = 0. - Colin Barker, Feb 08 2014
Except for the first term, positive values of x (or y) satisfying x^2 - 14*x*y + y^2 + 432 = 0. - Colin Barker, Feb 16 2014
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LINKS
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FORMULA
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a(n) = 4*a(n-1) - a(n-2).
G.f.: 3*x*(1-2*x)/(1-4*x+x^2). (End)
a(n) = 3*( (2+sqrt(3))*(2-sqrt(3))^n + (2-sqrt(3))*(2+sqrt(3))^n )/2. - Colin Barker, Oct 12 2015
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MATHEMATICA
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CoefficientList[Series[3(1-2x)/(1-4x+x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2014 *)
LinearRecurrence[{4, -1}, {3, 6}, 30] (* Harvey P. Dale, Dec 21 2022 *)
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PROG
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(Magma) I:=[3, 6]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 11 2014
(PARI) Vec(3*x*(1-2*x)/(1-4*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015
(SageMath) [3*chebyshev_T(n, 2) for n in range(41)] # G. C. Greubel, Oct 10 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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