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A075844
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Numbers n such that 11*n^2 + 4 is a square.
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3
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0, 6, 120, 2394, 47760, 952806, 19008360, 379214394, 7565279520, 150926376006, 3010962240600, 60068318435994, 1198355406479280, 23907039811149606, 476942440816512840, 9514941776519107194, 189821893089565631040
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OFFSET
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0,2
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COMMENTS
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Lim. n-> Inf. a(n)/a(n-1) = 10 + 3*sqrt(11).
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REFERENCES
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A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.
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LINKS
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FORMULA
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a(n) = ((10+3*sqrt(11))^n - (10-3*sqrt(11))^n) / sqrt(11).
a(n) = 20*a(n-1) - a(n-2).
G.f.: 6*x/(1 - 20*x + x^2).
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MAPLE
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seq(coeff(series(6*x/(1-20*x+x^2), x, n+1), x, n), n = 0..20); # G. C. Greubel, Dec 06 2019
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MATHEMATICA
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LinearRecurrence[{20, -1}, {0, 6}, 20] (* Harvey P. Dale, May 28 2012 *)
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PROG
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(PARI) my(x='x+O('x^20)); concat([0], Vec(6*x/(1-20*x+x^2))) \\ G. C. Greubel, Dec 06 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 20); [0] cat Coefficients(R!( 6*x/(1 - 20*x + x^2) )); // G. C. Greubel, Dec 06 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 6*x/(1-20*x+x^2) ).list()
(GAP) a:=[0, 6];; for n in [3..20] do a[n]:=20*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 06 2019
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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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