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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
OFFSET
1,1
COMMENTS
Alternatively, numbers of the form k*(4^n + 2^n + 1), where 2^(n-1) <= k < 2^n.
LINKS
Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's Theorem for Binary Powers, arXiv:1801.04483 [math.NT], 2018.
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
KEYWORD
nonn,base
AUTHOR
Jeffrey Shallit, Dec 29 2017
STATUS
approved