A188485 -id:A188485 - cp's OEIS frontend

A329825 Beatty sequence for (3+sqrt(17))/4.

1, 3, 5, 7, 8, 10, 12, 14, 16, 17, 19, 21, 23, 24, 26, 28, 30, 32, 33, 35, 37, 39, 40, 42, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 105, 106, 108, 110, 112, 113

Offset: 1

Clark Kimberling, Nov 22 2019

Let r = (3+sqrt(17))/4. Then (floor(n*r)) and (floor(n*r + r/2)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. The sequence (a(n) mod 2) of 0's and 1's has only two run-lengths: 4 and 5.
More generally, suppose that t > 0. There exists an irrational number r such that (floor(n*r)) and (floor(n*(r+t))) are a pair of Beatty sequences. Specifically, r = (2 - t + sqrt(t^2 + 4))/2, as in the Mathematica code below. See Comments at A182760.
************
Guide to related sequences:
t = 1:   A000201 and A001950 (Wythoff sequences), r = (1+sqrt(5))/2
t = 1/2: A329825 and A329826, r = (3 + sqrt(17))/4
t = 1/3: A329827 and A329828, r = (5 + sqrt(37))/6
t = 2/3: A329829 and A329830, r = (2 + sqrt(10))/3
t = 1/4: A329831 and A329832, r = (7 + sqrt(65))/8
t = 3/4: A329833 and A329834, r = (5 + sqrt(73))/8
t = 1/5: A329835 and A329836, r = (9 + sqrt(101))/10
t = 2/5: A329837 and A329838, r = (4 + sqrt(26))/5
t = 5/2: A329839 and A329840, r = (-1 + sqrt(41))/4
t = 3/5: A329841 and A329842, r = (7 + sqrt(109))/10
t = 5/3: A329843 and A329844, r = (1 + sqrt(61))/6
t = 5/4: A329847 and A329848, r = (3 + sqrt(89))/8
t = 4/5: A329845 and A329846, r = (3 + sqrt(29))/5
t = 6/5: A329923 and A329924, r = (2 + sqrt(34))/5
t = 8/5: A329925 and A329926, r = (1 + sqrt(41))/5
t = 2:   A001951 and A001952, r = sqrt(2)
t = 3:   A001961 and A004976, r = -1 + sqrt(5)
t = 4:   A001961 and A001962, r = -1 + sqrt(5)
t = 5:   A184522 and A184523, r = (-3 + sqrt(29))/2
t = 6:   A187396 and A187395, r = -2 + sqrt(10).
Starts to deviate from A059565 at a(73). - R. J. Mathar, Nov 26 2019
Sequences for t = 5/4, 4/5 and 3 corrected by Georg Fischer, Aug 22 2021

Clark Kimberling, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A188485, A329826 (complement), A182760.

t = 1/2; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]  (* A329825 *)
Table[Floor[s*n], {n, 1, 200}]  (* A329826 *)

a(n)=(sqrtint(17*n^2)+3*n)\4 \\ Charles R Greathouse IV, Jan 25 2022

a(n) = floor(r*n), where r = (3+sqrt(17))/4.

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.

A188934 Decimal expansion of (1+sqrt(17))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a (1/2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A (1/2)-extension rectangle matches the continued fraction [1,3,1,1,3,1,1,3,1,1,3,...] for the shape L/W=(1+sqrt(17))/4.  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 (1/2)-extension rectangle, 1 square is removed first, then 3 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(17))/4 is partitioned into an infinite collection of squares.
Conjecture: This number is an eigenvalue to infinitely many n*n submatrices of A191898, starting in the upper left corner, divided by the row index. For the first few characteristic polynomials see A260237 and A260238. - Mats Granvik, May 12 2016.

			1.2807764064044151374553524639935192562...

Cf. A188640, A188935.

Essentially the same as A188485.

r = 1/2; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
(* for the continued fraction *) ContinuedFraction[t, 120]
RealDigits[(1 + Sqrt@ 17)/4, 10, 111][[1]] (* Or *)
RealDigits[Exp@ ArcSinh[1/4], 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)

(sqrt(17)+1)/4 \\ Charles R Greathouse IV, May 12 2016

A285402 Positions of 1 in A285177; complement of A285401.

Original entry on oeis.org

3, 6, 9, 12, 13, 14, 15, 16, 19, 22, 25, 28, 31, 32, 33, 34, 35, 38, 41, 44, 47, 50, 51, 52, 53, 54, 57, 60, 63, 66, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 97, 98, 99, 100, 101, 104, 107, 110, 113, 116, 117, 118, 119

Offset: 1

Clark Kimberling, Apr 26 2017

Conjecture: a(n)/n -> (82 + sqrt(3))/47 = 1.781...

This conjecture is false. In fact, a(n)/n --> (5+sqrt(17))/(1+sqrt(17)) = 1.7807764... = A188485. See A285401. It follows in the same way as there, that a(n)/n --> 1/f1, where f1 is the frequency of 1's in A285177, and f1 can be computed using the Perron Frobenius theorem. - Michel Dekking, Feb 10 2021

			As a word, A285177 = 001001..., in which 0 is in positions 3,6,9,12,13,...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A188485, A285177, A285401, A285403.

s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 0, 1}}] &, {0}, 10] (* A285177 *)
Flatten[Position[s, 0]]  (* A285401 *)
Flatten[Position[s, 1]]  (* A285402 *)

A189966 Decimal expansion of (3+sqrt(33))/4, which has periodic continued fractions [2,5,2,1,2,5,2,1,...] and [3/2, 1, 3/2, 1, ...].

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2011

Let R denote a rectangle whose shape (i.e., length/width) is (3+sqrt(33))/4. This rectangle can be partitioned into squares in a manner that matches the continued fraction [2,5,2,1,2,5,2,1,2,5,2,1,...]. It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [3/2, 1, 3/2, 1, 3/2, ...]. For details, see A188635.

Apart from the first digit, the same as A188939. - R. J. Mathar, May 16 2011

			2.18614066163450716496265286705473232955506611449...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A188635, A188485.

(3+Sqrt(33))/4 // G. C. Greubel, Jan 12 2018

FromContinuedFraction[{3/2, 1, {3/2, 1}}]
ContinuedFraction[%, 25]  (* [2,5,2,1,2,5,2,1,...] *)
RealDigits[N[%%, 120]]  (* A189966 *)
N[%%%, 40]

(3+sqrt(33))/4 \\ G. C. Greubel, Jan 12 2018

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A329825 Beatty sequence for (3+sqrt(17))/4.

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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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A188934 Decimal expansion of (1+sqrt(17))/4.

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A285402 Positions of 1 in A285177; complement of A285401.

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A189966 Decimal expansion of (3+sqrt(33))/4, which has periodic continued fractions [2,5,2,1,2,5,2,1,...] and [3/2, 1, 3/2, 1, ...].

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