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A165225
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a(0)=1, a(1)=5, a(n) = 10*a(n-1) - 5*a(n-2) for n > 1.
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6
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1, 5, 45, 425, 4025, 38125, 361125, 3420625, 32400625, 306903125, 2907028125, 27535765625, 260822515625, 2470546328125, 23401350703125, 221660775390625, 2099601000390625, 19887706126953125, 188379056267578125
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OFFSET
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0,2
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COMMENTS
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Sum_{k=1..(m-1)/2} tan^(2n) (k*Pi/m) is an integer when m >= 3 is an odd integer (see AMM and Crux Mathematicorum links); twice this sequence is the particular case m = 5. - Bernard Schott, Apr 25 2022
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LINKS
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FORMULA
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Limit_{n->oo} a(n+1)/a(n) = 5 + 2*sqrt(5) = 9.47213595...
G.f.: (1-5x)/(1-10x+5x^2).
a(n) = ((5 - 2*sqrt(5))^n + (5 + 2*sqrt(5))^n)/2. - Klaus Brockhaus, Sep 25 2009
a(n) = (tan(Pi/5)^(2*n) + tan(2*Pi/5)^(2*n))/2 (Smeenk, 2009). - Amiram Eldar, Apr 03 2022
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MATHEMATICA
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LinearRecurrence[{10, -5}, {1, 5}, 30] (* Harvey P. Dale, Dec 23 2019 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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