A152422 -id:A152422 - cp's OEIS frontend

A375069 Decimal expansion of the sagitta of a regular hexagon with unit side length.

1, 3, 3, 9, 7, 4, 5, 9, 6, 2, 1, 5, 5, 6, 1, 3, 5, 3, 2, 3, 6, 2, 7, 6, 8, 2, 9, 2, 4, 7, 0, 6, 3, 8, 1, 6, 5, 2, 8, 5, 9, 7, 3, 7, 3, 0, 9, 4, 8, 0, 9, 6, 8, 5, 9, 7, 2, 0, 9, 6, 5, 1, 0, 2, 7, 4, 0, 3, 3, 4, 9, 1, 5, 4, 5, 5, 9, 9, 9, 8, 1, 4, 5, 9, 4, 2, 6, 9, 0, 6

Offset: 0

Paolo Xausa, Jul 30 2024

			0.133974596215561353236276829247063816528597373...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Sagitta
Index entries for algebraic numbers, degree 2.

Essentially the same as A334843.
Cf. A010527 (apothem), A104956 (area).
Cf. sagitta of other polygons with unit side length: A020769 (triangle), A174968 (square), A375068 (pentagon), A374972 (heptagon), A375070 (octagon), A375153 (9-gon), A375189 (10-gon), A375192 (11-gon), A375194 (12-gon).
Cf. A010527, A019913, A152422, A332133.

First[RealDigits[Tan[Pi/12]/2, 10, 100]]

tan(Pi/12)/2 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(4*x^2-8*x+1)[1] \\ Charles R Greathouse IV, Feb 04 2025

Equals tan(Pi/12)/2 = A019913/2.
Equals 1 - sqrt(3)/2 = 1 - A010527.
Equals A152422^2 = (1 - A332133)^2. - Hugo Pfoertner, Jul 30 2024
Equals A334843-1/2. - R. J. Mathar, Aug 02 2024

A332133 Decimal expansion of (1 + sqrt(3))/2, unique positive root of x^2 - x - 1/2.

Original entry on oeis.org

1, 3, 6, 6, 0, 2, 5, 4, 0, 3, 7, 8, 4, 4, 3, 8, 6, 4, 6, 7, 6, 3, 7, 2, 3, 1, 7, 0, 7, 5, 2, 9, 3, 6, 1, 8, 3, 4, 7, 1, 4, 0, 2, 6, 2, 6, 9, 0, 5, 1, 9, 0, 3, 1, 4, 0, 2, 7, 9, 0, 3, 4, 8, 9, 7, 2, 5, 9, 6, 6, 5, 0, 8, 4, 5, 4, 4, 0, 0, 0, 1, 8, 5, 4, 0, 5, 7, 3, 0, 9, 3, 3, 7, 8, 6, 2, 4, 2, 8, 7, 8, 3, 7, 8

Offset: 1

M. F. Hasler, Oct 29 2020

Also, max {a, b} where {a,b} is the unique solution of a + b = 1 and a^2 + b^2 = 2 (implying also ab = -1/2 and a^3 + b^3 = 5/2 without solving for a, b). See A332122 for a generalization to 3 variables {a, b, c}.

This is a non-integer element of the quadratic number field Q(sqrt(3)) with the given monic minimal polynomial. The other negative root is -(-1 + sqrt(3))/2 = - A152422. - Wolfdieter Lang, Aug 30 2022

			1.3660254037844386467637231707529361834714026269051903140279...

Index entries for algebraic numbers, degree 2

Cf. A152422 (this - 1 = (sqrt(3)-1)/2), A010527, A332122 (analog for 3rd degree).

Cf. A001622, A174968.

RealDigits[(1 + Sqrt[3])/2, 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

localprec(111); digits(solve(a=0,2,a^2-a-1/2)\.1^99)

polrootsreal(2*x^2-2*x-1)[2] \\ Charles R Greathouse IV, Jan 26 2023

Equals 1/2 + Sum_{n>=0} ((-1)^(n + 1)*binomial(2*n, n))/(2^(3*n + 1/2)*(2*n - 1)). - Antonio Graciá Llorente, Nov 11 2024

A173299 Numerators of fractions x^n + y^n, where x + y = 1 and x^2 + y^2 = 2.

Original entry on oeis.org

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

Offset: 1

J. Lowell, Feb 15 2010

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

			a(3) = 5 because x^3 + y^3 is 2.5 and 2.5 is 5/2.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A173300 (denominators).

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

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

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 *)

{ a(n) = numerator( 2 * polcoeff( lift( Mod((1+x)/2,x^2-3)^n ), 0) ) }

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

a(n) = numerator of ((1 + sqrt(3))/2)^n + ((1 - sqrt(3))/2)^n.

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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A375069 Decimal expansion of the sagitta of a regular hexagon with unit side length.

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A332133 Decimal expansion of (1 + sqrt(3))/2, unique positive root of x^2 - x - 1/2.

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A173299 Numerators of fractions x^n + y^n, where x + y = 1 and x^2 + y^2 = 2.

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