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A006061
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Star numbers (A003154) that are squares.
(Formerly M5385)
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8
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1, 121, 11881, 1164241, 114083761, 11179044361, 1095432263641, 107341182792481, 10518340481399521, 1030690025994360601, 100997104206965939401, 9896685522256667700721, 969774184076946468731281
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OFFSET
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1,2
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REFERENCES
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J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 121, p. 42, Ellipses, Paris 2008.
M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 22.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = denominator of kappa(sqrt(6)/A054320(n)) where kappa(x) is the sum of successive remainders by computing the Euclidean algorithm for (1, x). - Thomas Baruchel, Nov 29 2003
a(n) = 99*(a(n-1) - a(n-2)) + a(n-3).
a(n) = (5 - 2*sqrt(6))/8*(sqrt(3) + sqrt(2))^(4*n) + (5 + 2*sqrt(6))/8*(sqrt(3) - sqrt(2))^(4*n) - 1/4. (End)
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EXAMPLE
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a(2)=121 because this is the 2nd star number (A003154) that is a square.
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MAPLE
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Digits := 1000:q := seq(floor(evalf(( (5+2*sqrt(6))^n*(sqrt(6)-2)-(5-2*sqrt(6))^n*(sqrt(6)+2))^2/16)), n=1..100);
A006061:=-(1+22*z+z**2)/(z-1)/(z**2-98*z+1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
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MATHEMATICA
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CoefficientList[Series[(1+22*x+x^2)/((1-x)*(1-98*x+x^2)), {x, 0, 20}], x] (* or *) LinearRecurrence[{99, -99, 1}, {1, 121, 11881}, 20] (* G. C. Greubel, Jul 23 2019 *)
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PROG
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(PARI) my(x='x+O('x^20)); Vec((1+22*x+x^2)/((1-x)*(1-98*x+x^2))) \\ G. C. Greubel, Jul 23 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+22*x+x^2)/((1-x)*(1-98*x+x^2)) )); // G. C. Greubel, Jul 23 2019
(Sage) ((1+22*x+x^2)/((1-x)*(1-98*x+x^2))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jul 23 2019
(GAP) a:=[1, 121, 11881];; for n in [4..20] do a[n]:=99*a[n-1]-99*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jul 23 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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EXTENSIONS
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STATUS
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approved
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