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A028231
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From hexagons in a circle problem.
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4
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1, 22, 313, 4366, 60817, 847078, 11798281, 164328862, 2288805793, 31878952246, 444016525657, 6184352406958, 86136917171761, 1199732487997702, 16710117914796073, 232741918319147326, 3241676738553266497, 45150732421426583638, 628868577161418904441
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OFFSET
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0,2
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COMMENTS
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Given by the numerators of the convergents to the continued fraction [1,(1,2)^i,3,(1,2)^{i-1},1]. - Jeffrey Shallit, Dec 11 2017
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REFERENCES
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J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
T. Nagell, Des équations indéterminées x^2 + x + 1 = y^n et x^2 + x + 1 = 3*y^n, Norsk Mat. Forenings Skrifter, Ser. I, (1921).
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LINKS
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FORMULA
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a(n) = sqrt(3)*((2+sqrt(3))^(2*n+1) - (2-sqrt(3))^(2*n+1))/4 - 1/2 (see Kevin A. Broughan paper). - Michel Marcus, Jul 28 2012
a(n) = 15*a(n-1)-15*a(n-2)+a(n-3). G.f.: (1+7*x-2*x^2)/((1-x)*(1-14*x+x^2)). - conjectured by Colin Barker, Apr 10 2012; these follow easily from the formula.
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MAPLE
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f:= gfun:-rectoproc({a(n) = 15*a(n-1)-15*a(n-2)+a(n-3), a(0)=1, a(1)=22, a(2)=313}, a(n), remember):
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MATHEMATICA
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With[{k = Sqrt@ 3}, Simplify@ Array[k ((2 + k)^(2 # + 1) - (2 - k)^(2 # + 1))/4 - 1/2 &, 19, 0]] (* Michael De Vlieger, Dec 11 2017 *)
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PROG
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(PARI) a(n) = {w = quadgen(12); w*((2+w)^(2*n+1) - (2-w)^(2*n+1))/4 - 1/2; } /* Michel Marcus, Jul 28 2012 */
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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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