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A297405 Binary "cubes"; numbers whose binary representation consists of three consecutive identical blocks. 3
7, 42, 63, 292, 365, 438, 511, 2184, 2457, 2730, 3003, 3276, 3549, 3822, 4095, 16912, 17969, 19026, 20083, 21140, 22197, 23254, 24311, 25368, 26425, 27482, 28539, 29596, 30653, 31710, 32767, 133152, 137313, 141474, 145635, 149796, 153957, 158118, 162279, 166440, 170601, 174762, 178923, 183084, 187245 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Alternatively, numbers of the form k*(4^n + 2^n + 1), where 2^(n-1) <= k < 2^n.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's Theorem for Binary Powers, arXiv:1801.04483 [math.NT], 2018.

Index entries for sequences related to binary expansion of n

FORMULA

a(n) = n*(1 + 2^p + 4^p) with p = 1 + floor(log_2(n)). - Alois P. Heinz, Dec 29 2017

G.f.: (7*x + Sum_{n>=1} (4^n + 3*8^n + (2^n + 2*4^n - 3*8^n)*x)*x^(2^n))/(1-x)^2. - Robert Israel, Dec 31 2017

EXAMPLE

42 in base 2 is 101010, which consists of three copies of the block "10".

MAPLE

a:= n-> (p-> n*(1+2^p+4^p))(1+ilog2(n)):

seq(a(n), n=1..50); # Alois P. Heinz, Dec 29 2017

MATHEMATICA

bc[n_]:=FromDigits[Join[n, n, n], 2]; Flatten[Table[bc/@Select[Tuples[ {1, 0}, n], #[[1]] == 1&], {n, 6}]]//Union (* Harvey P. Dale, Oct 09 2021 *)

PROG

(Python)

def a(n): return int(bin(n)[2:]*3, 2)

print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Jul 04 2022

# Alternative:

def A297405(n):

p = n.bit_length()

return n * (1 + 2**p + 4**p)

print([A297405(n) for n in range(1, 46)]) # Peter Luschny, Jul 05 2022

(PARI) a(n) = n=binary(n); fromdigits(concat([n, n, n]) , 2) \\ Iain Fox, Jul 04 2022

CROSSREFS

Cf. A020330, which is the corresponding sequence for squares.

Subsequence of A121016.

Sequence in context: A188066 A225327 A102532 * A044109 A195320 A110451

Adjacent sequences: A297402 A297403 A297404 * A297406 A297407 A297408

KEYWORD

nonn,base

AUTHOR

Jeffrey Shallit, Dec 29 2017

STATUS

approved

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Last modified December 8 03:48 EST 2022. Contains 358672 sequences. (Running on oeis4.)