A246725 -id:A246725 - cp's OEIS frontend

A246724 Decimal expansion of r_2, the second smallest radius for which a compact packing of the plane exists, with disks of radius 1 and r_2.

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

Offset: 0

Jean-François Alcover, Sep 02 2014

Essentially the same digit sequence as A176053 and A020832. - R. J. Mathar, Sep 06 2014
This equals the ratio of the radius of the inner Soddy circle and the common radius of the three kissing circles. See A343235, also for links. - Wolfdieter Lang, Apr 19 2021
Previous comment is, together with A176053, the answer to the 1st problem proposed during the 4th Canadian Mathematical Olympiad in 1972. - Bernard Schott, Mar 16 2022

			0.154700538379251529018297561003914911295203502540253752...

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993 - Canadian Mathematical Society & Société Mathématique du Canada, Problem 1, 1972, page 37, 1993.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 62.
The IMO Compendium, Problem 1, 4th Canadian Mathematical Olympiad, 1972.
Samuel G. Moreno and Esther M. García, New infinite products of cosines and Viète-like formulae, Mathematics Magazine, Vol. 86, No. 1 (2013), pp. 15-25.
Bernard Schott, Soddy circles.
Index entries for algebraic numbers, degree 2.
Index to sequences related to Olympiads.

Cf. A246723 (r_1), A246725 (r_3), A246726 (r_4), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246729 (r_8), A246730 (r_9).

Cf. A176053, A343235.

RealDigits[(2*Sqrt[3] - 3)/3, 10, 103] // First

2/sqrt(3) - 1 \\ Charles R Greathouse IV, Feb 10 2025

Equals (2*sqrt(3) - 3)/3.
Equals A176053 - 2.
Equals -1 + sqrt(2) * sqrt(2-sqrt(2)) * sqrt(2-sqrt(2-sqrt(2))) * ... (Moreno and García, 2013). - Amiram Eldar, Jun 09 2022

A246723 Decimal expansion of r_1, the smallest radius for which a compact packing of the plane exists, with disks of radius 1 and r_1.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 02 2014

			0.101020514433643803605431850588217216068105038686659743...

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 62.
Gabriel Klambauer, Summation of Series, Amer. Math. Monthly, Vol. 87, No. 2 (Feb., 1980), pp. 128-130.
Wikipedia, Engel expansion
Index entries for algebraic numbers, degree 2

Cf. A246724 (r_2), A246725 (r_3), A246726 (r_4), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246729 (r_8), A246730 (r_9).

RealDigits[5 - 2*Sqrt[6], 10, 104] // First

Equals 5 - 2*sqrt(6).
Equals Sum_{k>=1} binomial(2*k,k)/((k+1) * 12^k). - Amiram Eldar, Oct 04 2021
Engel expansion of 5 - 2*sqrt(6) = 1/10 + 1/(10*98) + 1/(10*98*9602) + ..., where [10, 98, 9602, ...] = A135927. See Klambauer, p. 130. - Peter Bala, Feb 01 2022
Equals exp(-arccosh(5)). - Amiram Eldar, Jul 06 2023

A246726 Decimal expansion of r_4, the 4th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_4.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 02 2014

			0.3491981862085498764736232370456943152782620498437475...

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 73.
Index entries for algebraic numbers, degree 4.

Cf. A246723 (r_1), A246724 (r_2), A246725 (r_3), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246729 (r_8), A246730 (r_9).

Cf. A214726.

RealDigits[Root[x^4 - 28x^3 - 10x^2 + 4x + 1, x, 3], 10, 103] // First

3rd root of x^4 - 28x^3 - 10x^2 + 4x + 1.

Equals 1/(cosec(Pi/12)-1) = 1/(A214726 - 1). - Amiram Eldar, Mar 27 2022

A246727 Decimal expansion of r_5, the 5th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_5.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 02 2014

			0.3861061048585385422861371299469896994436146884586173...

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 73.
Index entries for algebraic numbers, degree 4.

Cf. A246723 (r_1), A246724 (r_2), A246725 (r_3), A246726 (r_4), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246729 (r_8), A246730 (r_9).

RealDigits[Root[9x^4 - 12x^3 - 26x^2 - 12x + 9, x, 1], 10, 104] // First

(1 + 2*sqrt(3) - 2*sqrt(1 + sqrt(3)))/3 \\ Charles R Greathouse IV, Feb 10 2025

1st root of 9x^4 - 12x^3 - 26x^2 - 12x + 9.

Equals (1 + 2*sqrt(3) - 2*sqrt(1 + sqrt(3)))/3. - Amiram Eldar, Mar 27 2022

A246728 Decimal expansion of r_7, the 7th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_7.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 02 2014

			0.53329641666031286760083809931809359409467606388528091693...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.
Index entries for algebraic numbers, degree 3.

Cf. A246723 (r_1), A246724 (r_2), A246725 (r_3), A246726 (r_4), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246729 (r_8), A246730 (r_9).

RealDigits[Root[8x^3 + 3x^2 - 2x - 1, x, 1], 10, 101] // First

polrootsreal(8*x^3+3*x^2-2*x-1)[1] \\ Charles R Greathouse IV, Feb 11 2025

1st root of 8x^3 + 3x^2 - 2x - 1.

A246729 Decimal expansion of r_8, the 8th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_8.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 02 2014

			0.5451510421225726875938077183373486963843555749734647529253568...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.
Index entries for algebraic numbers, degree 8.

Cf. A246723 (r_1), A246724 (r_2), A246725 (r_3), A246726 (r_4), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246730 (r_9).

RealDigits[Root[x^8 - 8x^7 - 44x^6 - 232x^5 - 482x^4 - 24x^3 + 388x^2 - 120x + 9, x, 3], 10, 101] // First

3rd root of x^8 - 8x^7 - 44x^6 - 232x^5 - 482x^4 - 24x^3 + 388x^2 - 120x + 9.

A246730 Decimal expansion of r_9, the 9th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_9.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 02 2014

			0.637555977231945793491317167739909596737570842457401874067...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.
Index entries for algebraic numbers, degree 4.

Cf. A246723 (r_1), A246724 (r_2), A246725 (r_3), A246726 (r_4), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246729 (r_8).

RealDigits[Root[x^4 - 10x^2 - 8x + 9, x, 1], 10, 102] // First

polrootsreal(x^4-10*x^2-8*x+9)[1] \\ Charles R Greathouse IV, Feb 11 2025

First root of x^4 - 10x^2 - 8x + 9.

A188485 Decimal expansion of (3+sqrt(17))/4, which has periodic continued fractions [1,1,3,1,1,3,1,1,3,...] and [3/2, 3, 3/2, 3, 3/2, ...].

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2011

Let R denote a rectangle whose shape (i.e., length/width) is (3+sqrt(17))/3.  This rectangle can be partitioned into squares in a manner that matches the continued fraction [1,1,3,1,1,3,1,1,3,...].  It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [3/2, 3, 3/2, 3, 3/2, ...].  For details, see A188635.
Apart from the second digit the same as A188934. - R. J. Mathar, May 16 2011
Equivalent to the infinite continued fraction with denominators [1; 2, 1, 2, 1, ...] and numerators [2, 1, 2, ...], also expressible as 1+2/(2+1/(1+2/(2+1/...))). - Matthew A. Niemiro, Dec 13 2019

			1.780776406404415137455352463993519256287...

J. S. Brauchart, P. D. Dragnev, E. B. Saff, An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge, arXiv preprint arXiv:1402.3367 [math-ph], 2014. See Footnote 8. - _N. J. A. Sloane_, Mar 26 2014
Index entries for algebraic numbers, degree 2.

Cf. A188934, A246725.

FromContinuedFraction[{3/2, 3, {3/2, 3}}]
ContinuedFraction[%, 25]  (* [1,1,3,1,1,3,1,1,3,...] *)
RealDigits[N[%%, 120]]  (* A188485 *)
N[%%%, 40]
RealDigits[(3+Sqrt[17])/4,10,120][[1]] (* Harvey P. Dale, May 09 2025 *)

cp's OEIS Frontend

A246724 Decimal expansion of r_2, the second smallest radius for which a compact packing of the plane exists, with disks of radius 1 and r_2.

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A246723 Decimal expansion of r_1, the smallest radius for which a compact packing of the plane exists, with disks of radius 1 and r_1.

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A246726 Decimal expansion of r_4, the 4th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_4.

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A246727 Decimal expansion of r_5, the 5th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_5.

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A246728 Decimal expansion of r_7, the 7th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_7.

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A246729 Decimal expansion of r_8, the 8th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_8.

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A246730 Decimal expansion of r_9, the 9th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_9.

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A188485 Decimal expansion of (3+sqrt(17))/4, which has periodic continued fractions [1,1,3,1,1,3,1,1,3,...] and [3/2, 3, 3/2, 3, 3/2, ...].

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