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A165806
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a(n) = 15n^2 + 3n + 1.
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10
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19, 67, 145, 253, 391, 559, 757, 985, 1243, 1531, 1849, 2197, 2575, 2983, 3421, 3889, 4387, 4915, 5473, 6061, 6679, 7327, 8005, 8713, 9451, 10219, 11017, 11845, 12703, 13591, 14509, 15457, 16435, 17443, 18481, 19549, 20647, 21775, 22933, 24121, 25339
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OFFSET
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1,1
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COMMENTS
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Polynomials f(x) have the following property: f(x + n*f(x)) is congruent to f(x); here n is an integer.
This can be proved by Taylor's theorem.
After rationalization of the denominator, the quotient q(n,x) = f(x + n*f(x))/f(x) consists of two parts:
a) a rational integer and b) an irrational part.
The present sequence is the integer part for f(x) = x^2 + 3x + 13 and x = sqrt(2), i.e., q(n,x) = a(n) + sqrt(2)*A045944(n).
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LINKS
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FORMULA
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G.f.: x*(19 + 10*x + x^2)/(1-x)^3. - R. J. Mathar, Sep 29 2009
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (15*x^2 + 18*x + 1)*exp(x). (End)
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EXAMPLE
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When we substitute sqrt(2) for x in the quadratic expression x^2 + 3x + 13 we get 15 + 3*sqrt(2).
sqrt(2) + (15 + 3*sqrt(2)) = (15 + 4*sqrt(2)). When this is substituted in f(x) we get 270 + 132*sqrt(2).
(270 + 132*sqrt(2))/(15+3*sqrt(2)) = 19 + 5*sqrt(2).
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {19, 67, 145}, 100] (* G. C. Greubel, Apr 08 2016 *)
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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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EXTENSIONS
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Definition simplified, sequence extended by R. J. Mathar, Sep 29 2009
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STATUS
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approved
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