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A020330
Numbers whose base-2 representation is the juxtaposition of two identical strings.
38
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
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.
Column k=0 of A246830, column k=1 of A246834.
Sequence in context: A233312 A330940 A351010 * A023861 A037345 A217278
KEYWORD
nonn,base,easy,look
AUTHOR
David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)
STATUS
approved