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 A173299 Numerators of fractions x^n + y^n, where x + y = 1 and x^2 + y^2 = 2. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS x and y are given by -A152422 and 1-A152422. - R. J. Mathar, Mar 01 2010 Letting f(n) = x^n + y^n, recurrence relation f(n) = f(n - 1) + f(n - 2)/2 implies a(n) / A173300(n) = A026150(n) / 2^(n - 1). - Nick Hobson, Jan 30 2024 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..200 FORMULA a(n) = numerator of ((1 + sqrt(3))/2)^n + ((1 - sqrt(3))/2)^n. EXAMPLE a(3) = 5 because x^3 + y^3 is 2.5 and 2.5 is 5/2. MAPLE 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 MATHEMATICA 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 *) PROG (PARI) { a(n) = numerator( 2 * polcoeff( lift( Mod((1+x)/2, x^2-3)^n ), 0) ) } (Magma) Z:=PolynomialRing(Integers()); N:=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 (Python) from fractions import Fraction def a173299_gen(a, b): while True: yield a.numerator b, a = b + Fraction(a, 2), b g = a173299_gen(1, 2) print([next(g) for _ in range(34)]) # Nick Hobson, Feb 20 2024 CROSSREFS Cf. A173300 (denominators). Sequence in context: A027038 A341322 A173929 * A097052 A102937 A045358 Adjacent sequences: A173296 A173297 A173298 * A173300 A173301 A173302 KEYWORD nonn,frac AUTHOR J. Lowell, Feb 15 2010 EXTENSIONS Formula, more terms, and PARI script from Max Alekseyev, Feb 24 2010 More terms from Klaus Brockhaus and R. J. Mathar, Mar 01 2010 STATUS approved

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Last modified May 21 08:56 EDT 2024. Contains 372733 sequences. (Running on oeis4.)