A140868 a(n) = floor(floor(n*alpha)*alpha) where alpha = 1+sqrt(2) = A014176.

4, 9, 16, 21, 28, 33, 38, 45, 50, 57, 62, 67, 74, 79, 86, 91, 98, 103, 108, 115, 120, 127, 132, 137, 144, 149, 156, 161, 168, 173, 178, 185, 190, 197, 202, 207, 214, 219, 226, 231, 236, 243, 248, 255, 260, 267, 272, 277, 284, 289, 296, 301, 306, 313, 318, 325, 330, 337, 342, 347, 354, 359, 366, 371, 376, 383

Offset: 1

Author

N. J. A. Sloane, Sep 04 2008

Keywords

• nonn
• easy

Comments

The sequence of first differences d = 5,7,5,7,5,5,7,... of this sequence, given by d(n) := a(n+1) - a(n), is equal to the fixed point of the morphism 5 -> 57, 7 -> 575. See Example 6 in my paper "Morphic words, Beatty sequences and integer images of the Fibonacci language". Modulo a change of alphabet, the sequence d occurs at many places in OEIS. See A006337, A159684, A080763, A276862, A276864. - Michel Dekking, Feb 18 2020

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Shiri Artstein-Avidan, Aviezri S. Fraenkel and Vera T. Sos, A two-parameter family of an extension of Beatty, Discr. Math. 308 (2008), 4578-4588.
• Shiri Artstein-avidan, Aviezri S. Fraenkel and Vera T. Sos, A two-parameter family of an extension of Beatty sequences, Discrete Math., 308 (2008), 4578-4588.
• M. Dekking, Morphic words, Beatty sequences and integer images of the Fibonacci language, Theoretical Computer Science 809, 407-417 (2020).

Crossrefs

Cf. A003151, A006337, A159684, A080763, A276862, A276864.

Programs

Magma

[Round(Floor((1+Sqrt(2))*Floor(n*(1+Sqrt(2))))): n in [1..100]]; // G. C. Greubel, Sep 27 2018

Maple

Digits := 200: a014176:= 1+sqrt(2) : A140868 := proc(n) global a014176 ; floor(a014176*floor(n*a014176)) ; end: for n from 1 to 100 do printf("%d,",A140868(n)); end: # R. J. Mathar, Sep 05 2008

Mathematica

With[{p = 1+Sqrt[2]}, Table[Floor[p*Floor[n*p]], {n, 1, 100}]] (* G. C. Greubel, Sep 27 2018 *)

PARI

vector(100, n, round(floor((1+sqrt(2))*floor(n*(1+sqrt(2)))))) \\ G. C. Greubel, Sep 27 2018

Python

from sympy import integer_nthroot
def A140868(n):
    f = lambda n: n+integer_nthroot(2*n**2,2)[0]
    return f(f(n)) # Chai Wah Wu, Mar 17 2021

Formula

a(n)= A003151(A003151(n)). - Michel Dekking, Feb 18 2020

Extensions

Corrected definition and extended by R. J. Mathar, Sep 05 2008