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A001535
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a(n) = (10n+1)*(10n+9).
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1
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9, 209, 609, 1209, 2009, 3009, 4209, 5609, 7209, 9009, 11009, 13209, 15609, 18209, 21009, 24009, 27209, 30609, 34209, 38009, 42009, 46209, 50609, 55209, 60009, 65009, 70209, 75609, 81209, 87009, 93009, 99209, 105609, 112209, 119009, 126009, 133209, 140609
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OFFSET
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0,1
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LINKS
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FORMULA
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a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=9, a(1)=209, a(2)=609. - Harvey P. Dale, Oct 15 2014
E.g.f.: (100*x^2 + 200*x + 9)*exp(x). - Robert Israel, Dec 17 2014
Sum_{n>=0} 1/a(n) = sqrt(5+2*sqrt(5))*Pi/80.
Sum_{n>=0} (-1)^n/a(n) = (sqrt(10+2*sqrt(5)) * log(cot(Pi/20)) + sqrt(10-2*sqrt(5)) * log(cot(3*Pi/20)))/80.
Product_{n>=0} (1 - 1/a(n)) = 2*phi*cos(sqrt(17)*Pi/10), where phi is the golden ratio (A001622).
Product_{n>=0} (1 + 1/a(n)) = 2*phi*cos(sqrt(15)*Pi/10). (End)
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MAPLE
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seq((10*n+1)*(10*n+9), n = 0 .. 100); # Robert Israel, Dec 17 2014
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MATHEMATICA
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Times@@@Table[10n+{1, 9}, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {9, 209, 609}, 40] (* Harvey P. Dale, Oct 15 2014 *)
CoefficientList[Series[(9 + 182 x + 9 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 17 2014 *)
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PROG
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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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STATUS
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approved
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