A187393 -id:A187393 - cp's OEIS frontend

A187394 a(n) = floor(s*n), where s = 4 - sqrt(8); complement of A187393.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 96, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113, 114, 115, 117

Offset: 1

Clark Kimberling, Mar 09 2011

nonn

A187393 and A187394 are the Beatty sequences based on r = 4 + sqrt(8) and s = 4 - sqrt(8); 1/r + 1/s = 1.

Let u = 1 + sqrt(2) and v = -1 + sqrt(2). Let U = {h*u, h >= 1} and V = {k*v, k >= 1}. Then A187393(n) is the position of n*u in the ordered union of U and V, and A187394 is the position of n*v. - Clark Kimberling, Oct 21 2014

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A187393.

[Floor (n*(4-Sqrt(8))): n in [0..100]]; // Vincenzo Librandi, Oct 23 2014

r=4+8^(1/2); s=4-8^(1/2);
Table[Floor[r*n],{n,1,80}]  (* A187393 *)
Table[Floor[s*n],{n,1,80}]  (* A187394 *)

from sympy import integer_nthroot
def A187394(n): return 4*n-1-integer_nthroot(8*n**2,2)[0] # Chai Wah Wu, Mar 16 2021

a(n) = floor(s*n), where s = 4 - sqrt(8).

A001952 A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).

Original entry on oeis.org

3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, 54, 58, 61, 64, 68, 71, 75, 78, 81, 85, 88, 92, 95, 99, 102, 105, 109, 112, 116, 119, 122, 126, 129, 133, 136, 139, 143, 146, 150, 153, 157, 160, 163, 167, 170, 174, 177, 180, 184, 187, 191, 194, 198

Offset: 1

N. J. A. Sloane

It appears that the distance between the a(n)-th triangular number and the nearest square is greater than floor(a(n)/2). - Ralf Stephan, Sep 14 2013

A080764(a(n)) = 0. - Reinhard Zumkeller, Jul 03 2015

Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5; MSRI Publications, Vol. 70 (2017), pages 101-153.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 77.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr. Pellian representatives, Fibonacci Quarterly, 10, issue 5, 1972, 449-488.
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8
A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=2).
Aviezri S. Fraenkel, On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications, Discrete Mathematics 224 (2000), pp. 273-279.
Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics, 179 (2014), 28-43. See Table 3.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Complement of A001951; equals A001951(n)+2*n.
A bisection of A094077.
Bisection: A187393, A342280.
Cf. A026250, A080764.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021

a001952 = floor . (* (sqrt 2 + 2)) . fromIntegral
-- Reinhard Zumkeller, Jul 03 2015

Table[Floor[n*(2 + Sqrt[2])], {n, 60}] (* Stefan Steinerberger, Apr 15 2006 *)
Array[Floor[#(2+Sqrt[2])]&,60] (* Harvey P. Dale, Dec 01 2015 *)

a(n)=2*n+sqrtint(2*n^2) \\ Charles R Greathouse IV, Jan 05 2016

from sympy import integer_nthroot
def A001952(n): return 2*n+integer_nthroot(2*n**2,2)[0] # Chai Wah Wu, Mar 16 2021

A342280 a(n) = A001952(2*n+1).

Original entry on oeis.org

3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, 78, 85, 92, 99, 105, 112, 119, 126, 133, 139, 146, 153, 160, 167, 174, 180, 187, 194, 201, 208, 215, 221, 228, 235, 242, 249, 256, 262, 269, 276, 283, 290, 297, 303, 310, 317, 324, 331, 338, 344, 351, 358, 365, 372

Offset: 0

N. J. A. Sloane, Mar 16 2021

nonn

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Bisection of A001952.

Cf. A187393.

from sympy import integer_nthroot
def A342280(n): return 4*n+2+integer_nthroot(8*n*(n+1)+2,2)[0] # Chai Wah Wu, Mar 16 2021

cp's OEIS Frontend

A187394 a(n) = floor(s*n), where s = 4 - sqrt(8); complement of A187393.

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A001952 A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).

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A342280 a(n) = A001952(2*n+1).

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