A246830 -id:A246830 - cp's OEIS frontend

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

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

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

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

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.

Subsequence of A121016.
Cf. A000051, A001576, A007088, A030308, A062383, A070939, A330157.
Column k=0 of A246830, column k=1 of A246834.

a020330 n = foldr (\d v -> 2 * v + d) 0 (bs ++ bs) where
   bs = a030308_row n
-- Reinhard Zumkeller, Feb 19 2013

[n+2*n*2^Floor(Log(2, n)): n in [1..50]]; // Vincenzo Librandi, Apr 05 2018

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

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 *)

a(n)=n+n<<#binary(n) \\ Charles R Greathouse IV, Mar 29 2013

is(n)=my(L=#binary(n)\2); n>>L==bitand(n,2^L-1) \\ Charles R Greathouse IV, Mar 29 2013

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

def A020330(n): return (n<Chai Wah Wu, Feb 28 2023

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

A080565 Binary expansion of n has form 11**...*1.

Original entry on oeis.org

3, 7, 13, 15, 25, 27, 29, 31, 49, 51, 53, 55, 57, 59, 61, 63, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233

Offset: 1

Benoit Cloitre, Feb 22 2003

nonn

If n>3 is in the sequence so are 2n-1 and 2n+1.

A004755 = union of A079946 and this sequence.

A diagonal of A246830.

a(n) = 2^floor(log[2](4*(n-1)))+2*n-1 for n>1, a(1)=3. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003

Equals 2 * A004760(n) + 1. - Ralf Stephan, Sep 16 2003

A246520 Largest number that can be written in binary as concatenation of (n - k) and (n + k), 0 <= k <= n.

Original entry on oeis.org

0, 3, 10, 20, 36, 45, 72, 104, 136, 153, 170, 208, 272, 336, 400, 464, 528, 561, 594, 627, 660, 693, 800, 928, 1056, 1184, 1312, 1440, 1568, 1696, 1824, 1952, 2080, 2145, 2210, 2275, 2340, 2405, 2470, 2535, 2600, 2665, 2730, 2880, 3136, 3392, 3648, 3904

Offset: 0

Reinhard Zumkeller, Sep 04 2014

Largest term in row n of A246830.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A246830, A007088, A246701 (first dfferences).

Haskell
```
a246520 = maximum . a246830_row
```

Table[Max[FromDigits[#,2]&/@Table[Join[IntegerDigits[n-k,2],IntegerDigits[ n+k,2]],{k,0,n}]],{n,0,50}] (* Harvey P. Dale, Oct 02 2018 *)

def A246520(n):
    return(max(int(bin(n-k)[2:]+bin(n+k)[2:],2) for k in range(n+1)))
# Chai Wah Wu, Sep 07 2014

A246831 a(n) is the concatenation of n and 3n in binary.

Original entry on oeis.org

0, 7, 22, 57, 76, 95, 210, 245, 280, 315, 350, 737, 804, 871, 938, 1005, 1072, 1139, 1206, 1273, 1340, 1407, 2882, 3013, 3144, 3275, 3406, 3537, 3668, 3799, 3930, 4061, 4192, 4323, 4454, 4585, 4716, 4847, 4978, 5109, 5240, 5371, 5502, 11137, 11396, 11655

Offset: 0

Alois P. Heinz, Sep 04 2014

			a(0) = (0|0)_2 = 0.
a(1) = (1|11)_2 = 7.
a(2) = (10|110)_2 = 22.
a(3) = (11|1001)_2 = 57.
a(4) = (100|1100)_2 = 76.
a(5) = (101|1111)_2 = 95.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A246830. Column k=3 of A246834.

f:= proc(i, j) local r, h, k; r:=0; h:=0; k:=j;
      while k>0 do r:=r+2^h*irem(k, 2, 'k'); h:=h+1 od; k:=i;
      while k>0 do r:=r+2^h*irem(k, 2, 'k'); h:=h+1 od; r
    end:
a:= n-> f(n, 3*n):
seq(a(n), n=0..60);

def A246831(n):
    return int(bin(n)[2:]+bin(3*n)[2:],2) # Chai Wah Wu, Sep 05 2014

a(n) = A246830(2n,n).

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

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A080565 Binary expansion of n has form 11**...*1.

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A246520 Largest number that can be written in binary as concatenation of (n - k) and (n + k), 0 <= k <= n.

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A246831 a(n) is the concatenation of n and 3n in binary.

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