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

3, 10, 15, 36, 45, 54, 63, 136, 153, 170, 187, 204, 221, 238, 255, 528, 561, 594, 627, 660, 693, 726, 759, 792, 825, 858, 891, 924, 957, 990, 1023, 2080, 2145, 2210, 2275, 2340, 2405, 2470, 2535, 2600, 2665, 2730, 2795, 2860, 2925, 2990, 3055, 3120, 3185, 3250

Offset

1,1

Comments

All differences are in union of A000051 and A001576. - Vladimir Shevelev, Dec 07 2013

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..8191

Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's Theorem for Binary Powers, Combinatorica, Vol. 39, No. 6 (2019), pp. 1335-1350, arXiv preprint, arXiv:1801.04483 [math.NT], 2018.

Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran, and Jeffrey Shallit, Lagrange's Theorem for Binary Squares, arXiv:1710.04247 [math.NT], 2017-2018.

Manfred Madritsch and Stephan Wagner, A central limit theorem for integer partitions, Monatshefte für Mathematik, Vol. 161, No. 1 (2010), pp. 85-114, alternative link.

Aayush Rajasekaran, Using Automata Theory to Solve Problems in Additive Number Theory, MS thesis, University of Waterloo, 2018.

Formula

a(n) = n + 2*n*2^floor(log_2(n)). - Ralf Stephan, Dec 07 2004

Sum_{n>=1} 1/a(n) = A330157. - Amiram Eldar, Oct 22 2020

a(n) = n * (2^A070939(n) + 1). - Jianing Song, Apr 10 2021

Example

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

Maple

a:= n-> (l-> Bits[Join]([l[], l[]]))(Bits[Split](n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 24 2024

Mathematica

Table[n + 2 n 2^Floor[Log[2, n]], {n, 50}] (* T. D. Noe, Dec 10 2013 *)
FromDigits[#, 2] & /@ (# <> # & /@ IntegerString[Range@100, 2]) (* Hans Rudolf Widmer, Aug 24 2024 *)

Prog

(Haskell)
a020330 n = foldr (\d v -> 2 * v + d) 0 (bs ++ bs) where
   bs = a030308_row n
-- Reinhard Zumkeller, Feb 19 2013
(PARI) a(n)=n+n<<#binary(n) \\ Charles R Greathouse IV, Mar 29 2013
(PARI) is(n)=my(L=#binary(n)\2); n>>L==bitand(n, 2^L-1) \\ Charles R Greathouse IV, Mar 29 2013
(Magma) [n+2*n*2^Floor(Log(2, n)): n in [1..50]]; // Vincenzo Librandi, Apr 05 2018
(Python)
def a(n): return int(bin(n)[2:]*2, 2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 10 2021
(Python)
def A020330(n): return (n<<n.bit_length())|n # Chai Wah Wu, Feb 28 2023

Crossrefs

Subsequence of A121016.

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

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

Sequence in context: A233312 A330940 A351010 * A023861 A037345 A217278

Adjacent sequences: A020327 A020328 A020329 * A020331 A020332 A020333

Keyword

nonn,base,easy,look

Author

David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)

Status

approved