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A140868
a(n) = floor(floor(n*alpha)*alpha) where alpha = 1+sqrt(2) = A014176.
2
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
1,1
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
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).
FORMULA
a(n)= A003151(A003151(n)). - Michel Dekking, Feb 18 2020
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 *)
PROG
(PARI) vector(100, n, round(floor((1+sqrt(2))*floor(n*(1+sqrt(2)))))) \\ G. C. Greubel, Sep 27 2018
(Magma) [Round(Floor((1+Sqrt(2))*Floor(n*(1+Sqrt(2))))): n in [1..100]]; // 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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Sep 04 2008
EXTENSIONS
Corrected definition and extended by R. J. Mathar, Sep 05 2008
STATUS
approved