A223139 -id:A223139 - cp's OEIS frontend

A356033 Decimal expansion of (-1 + sqrt(13))/6 = A223139/3.

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

Offset: 0

Wolfdieter Lang, Aug 29 2022

This constant r, an algebraic integer of the quadratic number field Q(13), is the positive root of its monic minimal polynomial x^2 + x/3 - 1/3. The negative root is -(1 + sqrt(13))/6 = -A209927/3 = -(A188943 - 1).

r^n = A052533(-n) + A006130(-(n+1))*r, for n >= 0, with A052533(-n) = 3*sqrt(-3)^(-n-2)*Snx(-n-2,1/sqrt(-3)), and A006130(-(n+1)) = sqrt(-3)^(-(n+1))*Snx(-(n+1), 1/sqrt(-3)), with the S-Chebyshev polynomials (see A049310), with S(-n, x) = -S(n-2, x), for n>=2, and S(-1, x) = 0. - Wolfdieter Lang, Nov 27 2023

			0.4342585459106648821865368779117493243752160956408743687850755...

Cf. A006130, A049310, A052533, A188943, A209927, A223139.

First[RealDigits[x/.N[Last[Solve[3x^2+x-1==0,x]],78]]] (* Stefano Spezia, Aug 29 2022 *)

r = (-1 + sqrt(13))/6 = A223139/3 = 1/A209927.

A122553 a(0)=1, a(n)=3 for n > 0.

Original entry on oeis.org

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Philippe Deléham, Sep 20 2006

Continued fraction for (sqrt(13) - 1)/2 = A223139.
Decimal expansion of 4/30. - Alonso del Arte, Aug 16 2012
4/3 is the volume of the regular octahedron inscribed in the unit-radius sphere. - Amiram Eldar, Jun 02 2023

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Springer, 2013, pp. 95-96, 224.

Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A118273 (cube), A339259 (regular icosahedron), A363437 (regular tetrahedron), A363438 (regular dodecahedron).

Cf. A223139.

RealDigits[4/3, 10, 105][[1]] (* Alonso del Arte, Aug 16 2012 *)
PadRight[{1},120,3] (* Harvey P. Dale, Jul 21 2023 *)

a(n)=(n>=0)+2*(n>0) \\ Jaume Oliver Lafont, Mar 26 2009

a(n) = 3 - 2*0^n.
G.f.: (1 + 2*x)/(1 - x).
Sum_{n >= 0} a(n)*10^(-n) = 4/3.
From Amiram Eldar, Jun 05 2021: (Start)
4/3 = Product_{k>=1} (1 + 1/2^(2^k)).
4/3 = Sum_{k>=0} binomial(2*k,k)/((k+2)*4^k). (End)
Sum_{k>0} 3*k/4^k = 4/3 [Nicole Oresme]. - Stefano Spezia, Jun 27 2024
K_{n>=3} n/(n-2) = 4/3 (see Clawson at p. 224). - Stefano Spezia, Jul 01 2024
E.g.f.: 3*exp(x) - 2. - Elmo R. Oliveira, Aug 05 2024

A209927 Decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))).

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Mar 17 2012

The number x given by the infinitely nested radical for n = 3 is such that x^2 = x + 3, bearing some similarity to the golden ratio phi with its property that phi^2 = phi + 1. Also, 3/x = x - 1.
The mentioned polynomial x^2 - x - 3 has the present number as positive root, and the negative one is -A223139. - Wolfdieter Lang, Aug 29 2022
It is the spectral radius of the bull-graph (see Seeger and Sossa, 2023). - Stefano Spezia, Sep 19 2023
c^n = A006130(n) + A006130(n-1) * d, where c = (1 + sqrt(13))/2 and d = (-1 + sqrt(13))/2. - Gary W. Adamson, Nov 25 2023
c^n = A052533(n) + A006130(n-1)*c, with A006130(-1) = 0. This is also valid for powers of 1/c = A356033, with A052533 and A006130 given there in terms of S-Chebyshev polynomials (A049310), used for negative indices. - Wolfdieter Lang, Nov 26 2023

			2.30277563773...

Alberto Seeger and David Sossa, Infinite families of connected graphs with equal spectral radius, Australas. J. Combin. 87 (2) (2023), 258-276. See p. 274.
Index entries for algebraic numbers, degree 2.

Cf. A157532 (continued fraction), A001622, A010470, A085550, A098316, A188943, A223139.

Cf. A006130, A049310, A052533, A209927.

Digits:=140:
evalf((sqrt(13)+1)/2);  # Alois P. Heinz, Sep 19 2023

RealDigits[(1 + Sqrt[13])/2, 10, 130][[1]]
RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[3, {n, 168}]], 10, 111][[1]] (* Robert G. Wilson v, Oct 02 2018 *)

(sqrt(13)+1)/2 \\ Altug Alkan, Oct 03 2018

Closed form: (sqrt(13) + 1)/2 = A098316-1 = A085550+2 = 3*(A188943-1).

A295330 Decimal expansion of sqrt(13)/2.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Nov 20 2017

In a regular hexagon inscribed in a circle of radius R the largest distance between any vertex and a midpoint of a side, after division of R, is sqrt(13)/2. The two smaller distance ratios are sqrt(7)/2 = A242703 and 1/2.
The regular period 6 continued fraction of sqrt(13)/2 is [1; 1, 4, 14, 4, 1, 2], and the convergents are given in A295331/A295332.
Essentially the same as A223139, A209927, A098316 and A085550. - R. J. Mathar, Nov 23 2017

			1.8027756377319946465596106337352479731256482869226231063552265281135834741465...

Index entries for algebraic numbers, degree 2.

Cf. A010470, A242703, A295331/A295332.

RealDigits[Sqrt@13/2, 10, 111][[1]] (* Robert G. Wilson v, Nov 20 2017 *)

sqrt(13)/2 \\ Michel Marcus, Nov 21 2017

A188943 Decimal expansion of (7 + sqrt(13))/6.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 14 2011

Decimal expansion of the shape (= length/width = (7+sqrt(13))/6) of the greater (7/3)-contraction rectangle.
See A188738 for an introduction to lesser and greater r-contraction rectangles, their shapes, and partitioning these rectangles into a sets of squares in a manner that matches the continued fractions of their shapes.
From Wolfdieter Lang, Aug 29 2022: (Start)
This constant t is an element of the quadratic number field Q(sqrt(13)) with (monic) polynomial x^2 - (7/3)*x + 1, and the negative root is -A188942.
The constant t - 1 = (1 + sqrt(13))/6 = A209927/3 has minimal polynomial x^2 - x/3 - 1/3, with negative root -(-1 + sqrt(13))/6 = -A223139/3 = -A356033.
(End)

			1.7675918792439982155198702112450826577085494289742...

Cf. A188738, A188942, A209927, A223139, A356033.

r = 7/3; t = (r + (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

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A356033 Decimal expansion of (-1 + sqrt(13))/6 = A223139/3.

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A122553 a(0)=1, a(n)=3 for n > 0.

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A209927 Decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))).

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A295330 Decimal expansion of sqrt(13)/2.

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A188943 Decimal expansion of (7 + sqrt(13))/6.

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