cp's OEIS Frontend

A297405 Binary "cubes"; numbers whose binary representation consists of three consecutive identical blocks.

%I A297405 #37 Jul 05 2022 16:02:40
%S A297405 7,42,63,292,365,438,511,2184,2457,2730,3003,3276,3549,3822,4095,
%T A297405 16912,17969,19026,20083,21140,22197,23254,24311,25368,26425,27482,
%U A297405 28539,29596,30653,31710,32767,133152,137313,141474,145635,149796,153957,158118,162279,166440,170601,174762,178923,183084,187245
%N A297405 Binary "cubes"; numbers whose binary representation consists of three consecutive identical blocks.
%C A297405 Alternatively, numbers of the form k*(4^n + 2^n + 1), where 2^(n-1) <= k < 2^n.
%H A297405 Robert Israel, <a href="/A297405/b297405.txt">Table of n, a(n) for n = 1..10000</a>
%H A297405 Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, <a href="https://arxiv.org/abs/1801.04483">Waring's Theorem for Binary Powers</a>, arXiv:1801.04483 [math.NT], 2018.
%H A297405 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A297405 a(n) = n*(1 + 2^p + 4^p) with p = 1 + floor(log_2(n)). - _Alois P. Heinz_, Dec 29 2017
%F A297405 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
%e A297405 42 in base 2 is 101010, which consists of three copies of the block "10".
%p A297405 a:= n-> (p-> n*(1+2^p+4^p))(1+ilog2(n)):
%p A297405 seq(a(n), n=1..50);  # _Alois P. Heinz_, Dec 29 2017
%t A297405 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 *)
%o A297405 (Python)
%o A297405 def a(n): return int(bin(n)[2:]*3, 2)
%o A297405 print([a(n) for n in range(1, 46)]) # _Michael S. Branicky_, Jul 04 2022
%o A297405 # Alternative:
%o A297405 def A297405(n):
%o A297405     p = n.bit_length()
%o A297405     return n * (1 + 2**p + 4**p)
%o A297405 print([A297405(n) for n in range(1, 46)])  # _Peter Luschny_, Jul 05 2022
%o A297405 (PARI) a(n) = n=binary(n); fromdigits(concat([n, n, n]) , 2) \\ _Iain Fox_, Jul 04 2022
%Y A297405 Cf. A020330, which is the corresponding sequence for squares.
%Y A297405 Subsequence of A121016.
%K A297405 nonn,base
%O A297405 1,1
%A A297405 _Jeffrey Shallit_, Dec 29 2017