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A173299
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Numerators of fractions x^n + y^n, where x + y = 1 and x^2 + y^2 = 2.
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4
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1, 2, 5, 7, 19, 13, 71, 97, 265, 181, 989, 1351, 3691, 2521, 13775, 18817, 51409, 35113, 191861, 262087, 716035, 489061, 2672279, 3650401, 9973081, 6811741, 37220045, 50843527, 138907099, 94875313, 518408351, 708158977, 1934726305, 1321442641
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OFFSET
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1,2
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COMMENTS
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x and y are given by -A152422 and 1-A152422. - R. J. Mathar, Mar 01 2010
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..200
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FORMULA
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a(n) = numerator of ((1 + sqrt(3))/2)^n + ((1 - sqrt(3))/2)^n.
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EXAMPLE
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a(3) = 5 because x^3 + y^3 is 2.5 and 2.5 is 5/2.
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MAPLE
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A173299 := proc(n) local x, y ; x := (1+sqrt(3))/2 ; y := (1-sqrt(3))/2 ; expand(x^n+y^n) ; numer(%) ; end proc: # R. J. Mathar, Mar 01 2010
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MATHEMATICA
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Module[{x=(1-Sqrt[3])/2, y}, y=1-x; Table[x^n+y^n, {n, 40}]]//Simplify// Numerator (* Harvey P. Dale, Aug 24 2019 *)
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PROG
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(PARI) { a(n) = numerator( 2 * polcoeff( lift( Mod((1+x)/2, x^2-3)^n ), 0) ) }
(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(2*x^2-2*x-1); S:=[ r^n+(1-r)^n: n in [1..34] ]; [ Numerator(RationalField()!S[j]): j in [1..#S] ]; // Klaus Brockhaus, Mar 02 2010
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CROSSREFS
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Cf. A173300 (denominators).
Sequence in context: A306918 A027038 A173929 * A097052 A102937 A045358
Adjacent sequences: A173296 A173297 A173298 * A173300 A173301 A173302
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KEYWORD
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nonn,frac
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AUTHOR
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J. Lowell, Feb 15 2010
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EXTENSIONS
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Formula, more terms, and PARI script from Max Alekseyev, Feb 24 2010
More terms from Klaus Brockhaus and R. J. Mathar, Mar 01 2010
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STATUS
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approved
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