A295333 -id:A295333 - cp's OEIS frontend

A020797 Decimal expansion of 1/sqrt(40).

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

Offset: 0

N. J. A. Sloane

With offset 1, decimal expansion of sqrt(5/2). - Eric Desbiaux, May 01 2008
sqrt(5/2) appears as a coordinate in a degree-5 integration formula on 13 points in the unit sphere [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011
With offset 2, decimal expansion of sqrt(250). - Michel Marcus, Nov 04 2013
From Wolfdieter Lang, Nov 21 2017: (Start)
The regular continued fraction of 1/sqrt(40) = 1/(2*sqrt(10)) is [0; 6, 3, repeat(12, 3)], and the convergents are given by A(n-1)/B(n-1), n >= 0, with A(-1) = 0, A(n-1) = A041067(n) and B(-1) = 1, B(n-1) = A041066(n).
The regular continued fraction of sqrt(5/2) = sqrt(10)/2 is [1; repeat(1, 1, 2)], and the convergents are given in A295333/A295334.
sqrt(10)/2 is one of the catheti of the rectangular triangle with hypotenuse sqrt(13)/2 = A295330 and the other cathetus sqrt(3)/2 = A010527. This can be constructed from a regular hexagon inscribed in a circle with a radius of 1 unit. If the vertex V_0 has coordinates (x, y) = (1, 0) and the midpoint M_4 = (0, -sqrt(3)/2) then the point L = (sqrt(10)/2, 0) is obtained as intersection of the x-axis and a circle around M_4 with radius taken from the distance between M_4 and V_1 = (1/2, sqrt(3)/2) which is sqrt(13)/2. (End)

			1/sqrt(40) = 0.15811388300841896659994467722163592668597775696626084134287...
sqrt(5/2) = 1.5811388300841896659994467722163592668597775696626084134287...
sqrt(250) = 15.811388300841896659994467722163592668597775696626084134287...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
A. H. Stroud, D. Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Index entries for algebraic numbers, degree 2.

Cf. A010467 (sqrt(10)), A010527, A010494 (sqrt(40)), A041067/A041066, A295330, A295333/A295334.

RealDigits[N[1/Sqrt[40],200]] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2010 *)

1/sqrt(40) \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(40*x^2-1)[2] \\ Charles R Greathouse IV, Feb 04 2025

Equals Re(sqrt(5*i)/10) = Im(sqrt(5*i)/10). - Karl V. Keller, Jr., Sep 01 2020

Equals A010467/20. - R. J. Mathar, Feb 23 2021

A295334 Denominators of continued fraction convergents to sqrt(10)/2 = sqrt(5/2) = A020797 + 1.

Original entry on oeis.org

1, 1, 2, 5, 7, 12, 31, 43, 74, 191, 265, 456, 1177, 1633, 2810, 7253, 10063, 17316, 44695, 62011, 106706, 275423, 382129, 657552, 1697233, 2354785, 4052018, 10458821, 14510839, 24969660, 64450159, 89419819, 153869978, 397159775, 551029753, 948189528, 2447408809, 3395598337, 5843007146

Offset: 0

Wolfdieter Lang, Nov 21 2017

The numerators are given in A295333. There details are given.

			For the first convergents see A295333.

Robert Israel, Table of n, a(n) for n = 0..3795
Index entries for linear recurrences with constant coefficients, signature (0, 0, 6, 0, 0, 1).

Cf. A020797, A295333.

numtheory:-cfrac(sqrt(5/2),100,'con'):
map(denom,con[1..-2]); # Robert Israel, Nov 22 2017

Denominator[Convergents[Sqrt[5/2], 50]] (* Wesley Ivan Hurt, Nov 21 2017 *)

G.f.: G(x) = (1 + x + 2*x^2 - x^3 + x^4)/(1 - 6*x^3 - x^6), For the derivation see A295333, but here the input of the recurrence is a(0) = 1, a(-1) = 0 (a(-2) = a(0) = 1). This leads here to G_0 = 1+ 2*x*G_2 + x*G_1, G_1 = G_0 + x*G_2, G_2 = G_1 + G_0 and the solution gives G(x).

a(n) = 6*a(n-3) + a(n-6), n >= 6, with inputs a(0)..a(5).

cp's OEIS Frontend

A020797 Decimal expansion of 1/sqrt(40).

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