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A120893
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a(n)=3*a(n-1) + 3*a(n-2) - a(n-3) ; a(0)=1,a(1)=1,a(2)=5.
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3
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1, 1, 5, 17, 65, 241, 901, 3361, 12545, 46817, 174725, 652081, 2433601, 9082321, 33895685, 126500417, 472105985, 1761923521, 6575588101, 24540428881, 91586127425, 341804080817, 1275630195845, 4760716702561, 17767236614401
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| For n>1,hypotenuse of primitive Pythagorean triangles having an angle nearing pi/3 for larger values of sides.[Complete triple (X,Y,Z),X<Y<Z is given by X=A120892(n),Y=A001353(n),Z=a(n) with recurrence relations X(i+1)=2*{a(i)-(-1)^i}-X(i-1) ; Y(i+1)=2*T(i)-T(i-1)-(-1)^i, where T(i)=Y(i)+a(i)] a(n)=2*A120892(n)-(-1)^n.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (3,3,-1).
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FORMULA
| Union of A103772 and A103974. a(n)=2*{2*a(n-1) + (-1)^n} - a(n-2) ; a(0)=1,a(1)=1.
a(n)=[(-1)^n+(2-sqrt(3))^n+(2+sqrt(3))^n]/3. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 24 2006
O.g.f: -(-1+2*x+x^2)/((1+x)*(x^2-4*x+1)). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 02 2007
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MAPLE
| a[0]:=1: a[1]:=1: a[2]:=5: for n from 3 to 40 do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3] od: seq(a[n], n=0..30); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 24 2006
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MATHEMATICA
| Transpose[NestList[Flatten[{Rest[#], 3Last[#]+3#[[2]]- First[#]}]&, {1, 1, 5}, 25]][[1]] (* or *)
CoefficientList[Series[(1-2 x-x^2)/(1-3 x-3 x^2+x^3), {x, 0, 25}], x] (* From Harvey P. Dale, Mar 27 2011 *)
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CROSSREFS
| Sequence in context: A149671 A062229 A196926 * A195531 A149672 A149673
Adjacent sequences: A120890 A120891 A120892 * A120894 A120895 A120896
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KEYWORD
| nonn
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AUTHOR
| Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 14 2006
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 24 2006
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