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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

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

Programs

Haskell

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

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 *)

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