A188934 -id:A188934 - cp's OEIS frontend

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.

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.

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 *)

A064651 a(n) = ceiling(a(n-1)/2) + a(n-2) with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 15, 20, 25, 33, 42, 54, 69, 89, 114, 146, 187, 240, 307, 394, 504, 646, 827, 1060, 1357, 1739, 2227, 2853, 3654, 4680, 5994, 7677, 9833, 12594, 16130, 20659, 26460, 33889, 43405, 55592, 71201, 91193, 116798, 149592

Offset: 0

Henry Bottomley, Oct 04 2001

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..9300

Cf. A064324, A064325.

Cf. A064650, A000045, A188934.

a064651 n = a064651_list !! n
a064651_list = 0 : 1 : zipWith (+)
   a064651_list (map (flip div 2 . (+ 1)) $ tail a064651_list)
-- Reinhard Zumkeller, Apr 30 2015

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==Ceiling[a[n-1]/2]+a[n-2]}, a, {n,50}] (* Harvey P. Dale, Aug 22 2012 *)
t = {0, 1}; Do[AppendTo[t, Ceiling[t[[-1]]/2] + t[[-2]]], {48}]; t (* T. D. Noe, Aug 22 2012 *)

a(n) = A064650(n) - 1.

Lim_{n->infinity} a(n)/a(n-1) = (1+sqrt(17))/4 = 1.2807764... = A188934.

A188935 Decimal expansion of (1+sqrt(37))/6.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a (1/3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A (1/3)-extension rectangle matches the continued fraction [1,5,1,1,5,1,1,5,1,1,5,...] for the shape L/W=(1+sqrt(37))/6. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the (1/3)-extension rectangle, 1 square is removed first, then 5 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(37))/6 is partitioned into an infinite collection of squares.

			1.1804604217163699481666140408670111770141616824644...

Cf. A188640, A188934.

RealDigits[(1 + Sqrt[37])/6, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)

Equals exp(arcsinh(1/6)). - Amiram Eldar, Jul 04 2023

cp's OEIS Frontend

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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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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A064651 a(n) = ceiling(a(n-1)/2) + a(n-2) with a(0)=0 and a(1)=1.

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A188935 Decimal expansion of (1+sqrt(37))/6.

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