A188934 -id:A188934

Search: a188934 -id:a188934

Displaying 1-4 of 4 results found.

page 1

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.

+10
8

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 (list; constant; graph; refs; listen; history; text; internal format)

	OFFSET	`0,1`
	COMMENTS	`Essentially the same digit sequence as A188934 and A188485. - R. J. Mathar, Sep 06 2014`
	LINKS	`Table of n, a(n) for n=0..102.` `Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.`
	FORMULA	`(sqrt(17) - 3)/4.`
	EXAMPLE	`0.2807764064044151374553524639935192562867998063434051...`
	MATHEMATICA	`RealDigits[(Sqrt[17] - 3)/4, 10, 103] // First`
	CROSSREFS	`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).`
	KEYWORD	`nonn,cons,easy`
	AUTHOR	`Jean-François Alcover, Sep 02 2014`
	STATUS	`approved`

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, ...].

+10
5

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 (list; constant; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`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`
	LINKS	`Table of n, a(n) for n=1..120.` `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`
	EXAMPLE	`1.780776406404415137455352463993519256287...`
	MATHEMATICA	`FromContinuedFraction[{3/2, 3, {3/2, 3}}]` `ContinuedFraction[%, 25] (* [1, 1, 3, 1, 1, 3, 1, 1, 3, ...] )` `RealDigits[N[%%, 120]] ( A188485 *)` `N[%%%, 40]`
	CROSSREFS	`Cf. A188934, A246725.`
	KEYWORD	`nonn,cons`
	AUTHOR	`Clark Kimberling, May 05 2011`
	STATUS	`approved`

A064651

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

+10
2

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,4`
	LINKS	`Reinhard Zumkeller, Table of n, a(n) for n = 0..9300`
	FORMULA	`a(n) = A064650(n) - 1.` `Lim_{n->infinity} a(n)/a(n-1) = (1+sqrt(17))/4 = 1.2807764... = A188934.`
	MATHEMATICA	`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 *)`
	PROG	`(Haskell)` `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`
	CROSSREFS	`Cf. A064324, A064325.` `Cf. A064650, A000045, A188934.`
	KEYWORD	`nonn`
	AUTHOR	`Henry Bottomley, Oct 04 2001`
	STATUS	`approved`

A188935

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

+10
2

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 (list; constant; graph; refs; listen; history; text; internal format)

	OFFSET	`1,3`
	COMMENTS	`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.`
	LINKS	`Table of n, a(n) for n=1..130.`
	FORMULA	`Equals exp(arcsinh(1/6)). - Amiram Eldar, Jul 04 2023`
	EXAMPLE	`1.1804604217163699481666140408670111770141616824644...`
	MATHEMATICA	`RealDigits[(1 + Sqrt[37])/6, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)`
	CROSSREFS	`Cf. A188640, A188934.`
	KEYWORD	`nonn,cons`
	AUTHOR	`Clark Kimberling, Apr 13 2011`
	STATUS	`approved`

page 1

Search completed in 0.005 seconds