A003152 A Beatty sequence: a(n) = floor(n*(1+1/sqrt(2))).

1, 3, 5, 6, 8, 10, 11, 13, 15, 17, 18, 20, 22, 23, 25, 27, 29, 30, 32, 34, 35, 37, 39, 40, 42, 44, 46, 47, 49, 51, 52, 54, 56, 58, 59, 61, 63, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 81, 83, 85, 87, 88, 90, 92, 93, 95, 97, 99, 100, 102, 104, 105, 107, 109, 110, 112, 114, 116

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Numbers with an even number of trailing 0's in their minimal representation in terms of the positive Pell numbers (A317204). - Amiram Eldar, Mar 16 2022

The indices of the squares in the sequence of squares and twice squares: A028982(a(n)) = n^2. - Amiram Eldar, Apr 13 2025

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Leonard Carlitz, Richard Scoville, and Verner E. Hoggatt, Jr., Pellian representations, Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.
• Joshua N. Cooper and Alexander W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, J. Int. Seq., Vol. 16 (2013), Article 13.1.8; preprint, 2012.
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
• Index entries for sequences related to Beatty sequences.

Crossrefs

Complement of A003151.

Cf. A028982, A109250, A317204.

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

Bisections: A001952, A001954.

Programs

Magma

[Floor(n*(1+1/Sqrt(2))): n in [1..70]]; // Vincenzo Librandi, Dec 26 2015

Maple

Digits := 100: t := evalf(1+sin(Pi/4)): A:= n->floor(t*n): seq(floor((t*n)),n=1..68); # Zerinvary Lajos, Mar 27 2009

Mathematica

Table[Floor[n (1 + 1/Sqrt[2])], {n, 70}] (* Vincenzo Librandi, Dec 26 2015 *)

PARI

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

Python

from math import isqrt
def A003152(n): return n+isqrt(n**2>>1) # Chai Wah Wu, May 24 2025