A246723 -id:A246723 - 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

A246725 Decimal expansion of r_3, the third smallest radius for which a compact packing of the plane exists, with disks of radius 1 and r_3.

Original entry on oeis.org

2, 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

Offset: 0

Jean-François Alcover, Sep 02 2014

Essentially the same digit sequence as A188934 and A188485. - R. J. Mathar, Sep 06 2014

			0.2807764064044151374553524639935192562867998063434051...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.
Index entries for algebraic numbers, degree 2.

Cf. A246723 (r_1), A246724 (r_2), A246726 (r_4), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246729 (r_8), A246730 (r_9).

RealDigits[(Sqrt[17] - 3)/4, 10, 103] // First

(sqrt(17)-3)/4 \\ Charles R Greathouse IV, Feb 10 2025

(sqrt(17) - 3)/4.

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.

A135927 a(n) = a(n-1)^2 - 2 with a(1) = 10.

Original entry on oeis.org

10, 98, 9602, 92198402, 8500545331353602, 72259270930397519221389558374402, 5221402235392591963136699520829303150191924374488750728808857602

Offset: 1

Ant King, Dec 07 2007

This is the Lucas-Lehmer sequence with starting value u(1) = 10 and the position of the zeros when it is reduced mod(2^p - 1) also gives the position of the Mersenne primes. As we have started with n = 1, these will occupy the (p - 1)th positions in the sequence. For example, the first 12 terms mod(2^13 - 1) are 10, 98, 1411, 506, 2113, 672, 1077, 4996, 2037, 4721, 128, 0 and hence 8191 is a Mersenne prime. The radicals in the above closed forms are the solutions to x^2 - 10x + 1 = 0.

			a(4) = 2*cosh(2^3*log(5 + 2*sqrt(6))) = 92198402.

Gabriel Klambauer, Summation of Series, Amer. Math. Monthly, Vol. 87, No. 2 (Feb., 1980), pp. 128-130.
Raphael M. Robinson, Mersenne and Fermat Numbers, Proceedings of the American Mathematical Society, Vol. 5, No. 5. (October 1954), pp. 842-846.
E. L. Roettger and H. C. Williams, Some Remarks Concerning the Lucas-Lehmer Primality Test, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.5. See p. 3.
Wikipedia, Engel expansion

Cf. A000668, A000043, A003010, A095847, A001566, A135928, A246723.

a[1] = 10; a[n_] := a[n] = a[n - 1]^2 - 2; a[#]&/@Range[7]

A135927 = [10]
for n in range(1, 8): A135927.append(A135927[-1]**2-2)
print(A135927) # Karl-Heinz Hofmann, Feb 01 2022

a(n) = 2*cosh(2^(n-1)*log(5 + 2*sqrt(6))) = exp(2^(n-1)*log(5 + 2*sqrt(6))) + exp(2^(n-1)*log(5 - 2*sqrt(6))) = (5 + 2*sqrt(6))^(2^(n-1)) + (5 - 2*sqrt(6))^(2^(n-1)) = ceiling(exp(2^(n-1)*log(5 + 2*sqrt(6)))) = ceiling((5 + 2*sqrt(6))^(2^(n-1))).
From Peter Bala, Feb 01 2022: (Start)
Product_{n >= 1} (1 + 2/a(n)) = (1/2)*sqrt(6); Product_{n >= 1} (1 - 1/a(n)) = (4/11)*sqrt(6).
Engel expansion of 5 - sqrt(24) = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + .... See Klambauer, p. 130. (End)

A340616 Decimal expansion of sqrt(3)-sqrt(2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 14 2021

From Bernard Schott, Jan 16 2021: (Start)
Equals the smallest positive root of x^4 - 10*x^2 + 1 (minimal polynomial).
An approximation to 1/Pi (see corresponding comment in A135611). (End)

			0.317837245195782244725...

Cf. A002193 (sqrt(2)), A002194 (sqrt(3)), A135611 (sqrt(2) + sqrt(3)), A246723 (5-2*sqrt(6)).

Cf. A172264 (Beatty sequence).

Maple
```
sqrt(3)-sqrt(2) ; evalf(%) ;
```

RealDigits[Sqrt[3] - Sqrt[2], 10, 100][[1]] (* Wesley Ivan Hurt, Jan 14 2021 *)

sqrt(3)-sqrt(2) \\ Michel Marcus, Jan 15 2021

Equals A002194 - A002193.
Equals A135611 - A010466.
Equals 1/A135611. - Bernard Schott, Jan 15 2021
Equals sqrt(A246723). - Kevin Ryde, Jan 15 2021

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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A246725 Decimal expansion of r_3, the third smallest radius for which a compact packing of the plane exists, with disks of radius 1 and r_3.

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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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A135927 a(n) = a(n-1)^2 - 2 with a(1) = 10.