|
| |
|
|
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
|
|
|
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
|
|
|
PROG
|
(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
|
|
|
CROSSREFS
|
Cf. A173300 (denominators).
Sequence in context: A306918 A027038 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
|
| |
|
|
