A020330 - OEIS

%I #92 Aug 24 2024 17:15:56

%S 3,10,15,36,45,54,63,136,153,170,187,204,221,238,255,528,561,594,627,

%T 660,693,726,759,792,825,858,891,924,957,990,1023,2080,2145,2210,2275,

%U 2340,2405,2470,2535,2600,2665,2730,2795,2860,2925,2990,3055,3120,3185,3250

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

%C All differences are in union of A000051 and A001576. - _Vladimir Shevelev_, Dec 07 2013

%H Reinhard Zumkeller, <a href="/A020330/b020330.txt">Table of n, a(n) for n = 1..8191</a>

%H 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 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 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 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 a(n) = n + 2*n*2^floor(log_2(n)). - _Ralf Stephan_, Dec 07 2004

%F Sum_{n>=1} 1/a(n) = A330157. - _Amiram Eldar_, Oct 22 2020

%F a(n) = n * (2^A070939(n) + 1). - _Jianing Song_, Apr 10 2021

%e 36 is a term because 36 = 100100_2, which is 100 followed by 100.

%p a:= n-> (l-> Bits[Join]([l[],l[]]))(Bits[Split](n)):

%p seq(a(n), n=1..50);  # _Alois P. Heinz_, Aug 24 2024

%t Table[n + 2 n 2^Floor[Log[2, n]], {n, 50}] (* _T. D. Noe_, Dec 10 2013 *)

%t FromDigits[#, 2] & /@ (# <> # & /@ IntegerString[Range@100, 2]) (* _Hans Rudolf Widmer_, Aug 24 2024 *)

%o (Haskell)

%o a020330 n = foldr (\d v -> 2 * v + d) 0 (bs ++ bs) where

%o    bs = a030308_row n

%o -- _Reinhard Zumkeller_, Feb 19 2013

%o (PARI) a(n)=n+n<<#binary(n) \\ _Charles R Greathouse IV_, Mar 29 2013

%o (PARI) is(n)=my(L=#binary(n)\2); n>>L==bitand(n,2^L-1) \\ _Charles R Greathouse IV_, Mar 29 2013

%o (Magma) [n+2*n*2^Floor(Log(2, n)): n in [1..50]]; // _Vincenzo Librandi_, Apr 05 2018

%o (Python)

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

%o print([a(n) for n in range(1, 51)]) # _Michael S. Branicky_, Mar 10 2021

%o (Python)

%o def A020330(n): return (n<<n.bit_length())|n # _Chai Wah Wu_, Feb 28 2023

%Y Subsequence of A121016.

%Y Cf. A000051, A001576, A007088, A030308, A062383, A070939, A330157.

%Y Column k=0 of A246830, column k=1 of A246834.

%K nonn,base,easy,look

%O 1,1

%A _David W. Wilson_, Melia Aldridge (ma38(AT)spruce.evansville.edu)