cp's OEIS Frontend

A020330 Numbers whose base-2 representation is the juxtaposition of two identical strings.

%I A020330 #92 Aug 24 2024 17:15:56
%S A020330 3,10,15,36,45,54,63,136,153,170,187,204,221,238,255,528,561,594,627,
%T A020330 660,693,726,759,792,825,858,891,924,957,990,1023,2080,2145,2210,2275,
%U A020330 2340,2405,2470,2535,2600,2665,2730,2795,2860,2925,2990,3055,3120,3185,3250
%N A020330 Numbers whose base-2 representation is the juxtaposition of two identical strings.
%C A020330 All differences are in union of A000051 and A001576. - _Vladimir Shevelev_, Dec 07 2013
%H A020330 Reinhard Zumkeller, <a href="/A020330/b020330.txt">Table of n, a(n) for n = 1..8191</a>
%H A020330 Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, <a href="https://doi.org/10.1007/s00493-019-3933-3">Waring's Theorem for Binary Powers</a>, Combinatorica, Vol. 39, No. 6 (2019), pp. 1335-1350, <a href="https://arxiv.org/abs/1801.04483">arXiv preprint</a>, arXiv:1801.04483 [math.NT], 2018.
%H A020330 Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran, and Jeffrey Shallit, <a href="https://arxiv.org/abs/1710.04247">Lagrange's Theorem for Binary Squares</a>, arXiv:1710.04247 [math.NT], 2017-2018.
%H A020330 Manfred Madritsch and Stephan Wagner, <a href="https://doi.org/10.1007/s00605-009-0126-y">A central limit theorem for integer partitions</a>, Monatshefte für Mathematik, Vol. 161, No. 1 (2010), pp. 85-114, <a href="https://www.researchgate.net/publication/225845584_A_central_limit_theorem_for_integer_partitions">alternative link</a>.
%H A020330 Aayush Rajasekaran, <a href="https://uwspace.uwaterloo.ca/handle/10012/13202">Using Automata Theory to Solve Problems in Additive Number Theory</a>, MS thesis, University of Waterloo, 2018.
%F A020330 a(n) = n + 2*n*2^floor(log_2(n)). - _Ralf Stephan_, Dec 07 2004
%F A020330 Sum_{n>=1} 1/a(n) = A330157. - _Amiram Eldar_, Oct 22 2020
%F A020330 a(n) = n * (2^A070939(n) + 1). - _Jianing Song_, Apr 10 2021
%e A020330 36 is a term because 36 = 100100_2, which is 100 followed by 100.
%p A020330 a:= n-> (l-> Bits[Join]([l[],l[]]))(Bits[Split](n)):
%p A020330 seq(a(n), n=1..50);  # _Alois P. Heinz_, Aug 24 2024
%t A020330 Table[n + 2 n 2^Floor[Log[2, n]], {n, 50}] (* _T. D. Noe_, Dec 10 2013 *)
%t A020330 FromDigits[#, 2] & /@ (# <> # & /@ IntegerString[Range@100, 2]) (* _Hans Rudolf Widmer_, Aug 24 2024 *)
%o A020330 (Haskell)
%o A020330 a020330 n = foldr (\d v -> 2 * v + d) 0 (bs ++ bs) where
%o A020330    bs = a030308_row n
%o A020330 -- _Reinhard Zumkeller_, Feb 19 2013
%o A020330 (PARI) a(n)=n+n<<#binary(n) \\ _Charles R Greathouse IV_, Mar 29 2013
%o A020330 (PARI) is(n)=my(L=#binary(n)\2); n>>L==bitand(n,2^L-1) \\ _Charles R Greathouse IV_, Mar 29 2013
%o A020330 (Magma) [n+2*n*2^Floor(Log(2, n)): n in [1..50]]; // _Vincenzo Librandi_, Apr 05 2018
%o A020330 (Python)
%o A020330 def a(n): return int(bin(n)[2:]*2, 2)
%o A020330 print([a(n) for n in range(1, 51)]) # _Michael S. Branicky_, Mar 10 2021
%o A020330 (Python)
%o A020330 def A020330(n): return (n<<n.bit_length())|n # _Chai Wah Wu_, Feb 28 2023
%Y A020330 Subsequence of A121016.
%Y A020330 Cf. A000051, A001576, A007088, A030308, A062383, A070939, A330157.
%Y A020330 Column k=0 of A246830, column k=1 of A246834.
%K A020330 nonn,base,easy,look
%O A020330 1,1
%A A020330 _David W. Wilson_, Melia Aldridge (ma38(AT)spruce.evansville.edu)