A001952 A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))). (Formerly M2534 N1001)

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

1,1

Comments

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

References

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).

Links

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

Mathematica

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 *)

Prog

(Haskell)
a001952 = floor . (* (sqrt 2 + 2)) . fromIntegral
-- Reinhard Zumkeller, Jul 03 2015
(PARI) a(n)=2*n+sqrtint(2*n^2) \\ Charles R Greathouse IV, Jan 05 2016
(Python)
from sympy import integer_nthroot
def A001952(n): return 2*n+integer_nthroot(2*n**2, 2)[0] # Chai Wah Wu, Mar 16 2021

Crossrefs

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

Sequence in context: A190007 A310052 A310053 * A189795 A145383 A258834

Adjacent sequences: A001949 A001950 A001951 * A001953 A001954 A001955

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved