A033015 -id:A033015 - cp's OEIS frontend

A033010 Numbers each of whose runs of digits in base 12 has length 2.

13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 1872, 1898, 1911, 1924, 1937, 1950, 1963, 1976, 1989, 2002, 2015, 3744, 3757, 3783, 3796, 3809, 3822, 3835, 3848, 3861, 3874, 3887, 5616, 5629, 5642, 5668, 5681, 5694, 5707, 5720, 5733, 5746, 5759, 7488, 7501

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Numbers without repeating adjacent digits for which all digits are divisible by 13, in base 144. Consequently there are 11^n n-digit members of this sequence (base 144) and so (11^(n+1)-1)/10 members of this sequence below 144^n. - Charles R Greathouse IV, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1400

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 12]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

from sympy.ntheory import digits
from itertools import groupby
def ok(n):
  return all(len(list(g))==2 for k, g in groupby(digits(n, 12)[1:]))
print(list(filter(ok, range(1, 7502)))) # Michael S. Branicky, Apr 27 2021

a(n) = 13*A043316(n) (= 13*n for n < 12). - M. F. Hasler, Feb 02 2014

A222813 Numbers whose binary representation is palindromic and in which all runs of 0's and 1's have length at least 2.

Original entry on oeis.org

3, 7, 15, 31, 51, 63, 99, 127, 195, 231, 255, 387, 455, 511, 771, 819, 903, 975, 1023, 1539, 1651, 1799, 1935, 2047, 3075, 3171, 3315, 3591, 3687, 3855, 3999, 4095, 6147, 6371, 6643, 7175, 7399, 7695, 7967, 8191, 12291, 12483, 12771, 13107, 13299, 14343, 14535, 14823, 15375, 15567, 15903, 16191, 16383, 24579

Offset: 1

N. J. A. Sloane, Mar 11 2013

These are the decimal representations of A061851 read as base-2 numbers.

The terms with an odd number L = 2k-1 of bits, i.e., 2^(L-1) < a(n) < 2^L, are given by the terms of A033015 with length k, shifted k-1 digits to the left and 'OR'ed with the binary reversal of the term. Terms with an even number L = 2k of digits are given as m*2^k + (binary reversal of m) where m runs over the k-bit terms from A033015 and the k-1 bit terms with the last bit negated appended). This explains the FORMULA for the number of terms of given size. - M. F. Hasler, Oct 17 2022

			51 (base 10) = 110011 (base 2), which is a palindrome and has three runs all of length 2.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 608 terms from N. J. A. Sloane)

Cf. A061851.

Cf. A006995 (binary palindromes), A033015 (no isolated binary digit), A028897 ("rebase" 10 -> 2).

brpalQ[n_]:=Module[{idn2=IntegerDigits[n,2]},idn2==Reverse[idn2] && Min[ Length/@ Split[idn2]]>1]; Select[Range[25000],brpalQ] (* Harvey P. Dale, May 21 2014 *)

is(n)=is_A033015(n)&&Vecrev(n=binary(n))==n \\ M. F. Hasler, Oct 06 2022

{A222813_row(n, s=A033015_row(n\/2))=apply(A030101, if(n%2, s\2, n>2, s=setunion([k*2+1-k%2|k<-A033015_row(n\2-1)],s), s=[1]))+s<<(n\2)} \\ Terms with n bits, i.e. between 2^(n-1) and 2^n. - M. F. Hasler, Oct 17 2022

From M. F. Hasler, Oct 06 2022: (Start)
Intersection of A006995 and A033015: binary palindromes with no isolated digit.
There are A000045(A004526(k)) = Fibonacci(floor(k/2)) terms between 2^(k-1) and 2^k.
a(n) = A028897(A061851(n)), where A028897 = convert binary to decimal. (End)

A355280 Binary numbers (digits in {0, 1}) with no run of digits with length < 2.

Original entry on oeis.org

11, 111, 1100, 1111, 11000, 11100, 11111, 110000, 110011, 111000, 111100, 111111, 1100000, 1100011, 1100111, 1110000, 1110011, 1111000, 1111100, 1111111, 11000000, 11000011, 11000111, 11001100, 11001111, 11100000, 11100011, 11100111, 11110000, 11110011, 11111000, 11111100, 11111111

Offset: 1

M. F. Hasler, Oct 17 2022

This is the binary representation of the terms in A033015.

The sequence can be seen as a table where row r contains the terms with r digits. Then row r+1 is obtained by from the terms of row r by duplicating their last digit, and from those of row r-1 by appending twice the 1's complement of their last digit. This yields the row lengths given in FORMULA.

			There can't be a terms with only 1 digit, so the smallest term is a(1) = 11.
The only 3-digit term is a(2) = 111, since in 100 the digit 1 is alone, and in 101 and 110 the digit 0 is alone.
With four digits we must have either no or two digits 0 and they must be at the end (to avoid isolated '1's), i.e., a(3) = 1100 and a(4) = 1111.

Robert Israel, Table of n, a(n) for n = 1..10945

Cf. A033015 (the same terms converted from base 2 to base 10),

Subsequence of A007088 (the binary numbers); A000042 (numbers in base 1) = A002275 \ {0} (repunits) are subsequences; A061851 is the subsequence of palindromes.

F:= proc(d) option remember;
   local R,i,j, x0;
   R:= NULL;
   for i from d-2 to 2 by -1 do
     x0:= (10^d - 10^i)/9;
     for j from i-2 to 0 by -1 do
        R:= R, op(map(t -> t + x0, procname(j)))
     od
   od;
   sort([R, (10^d-1)/9])
end proc:
F(0):= [0]; F(1):= [];
seq(op(F[i]),i=2..9); # Robert Israel, May 12 2025

{is_A355280(n,d=digits(n))=vecmax(d)==1 && is_A033015(fromdigits(d,2))}
A355280(n)=A007088(A033015(n))
concat(apply( {A355280_row(n)=if(n>2, setunion([x*10+x%10|x<-A355280_row(n-1)],[x*100+11*(1-x%10)|x<-A355280_row(n-2)]), n>1, [11],[])}, [1..8])) \\ "Row" of n-digit terms. For (very) large n one should implement memoization instead of this naive recursion.

def A355280_row(n): return [] if n<2 else [11] if n==2 else sorted(
    [x*10+x%10 for x in A355280_row(n-1)] +
    [x*100+11-x%10*11 for x in A355280_row(n-2)]) # M. F. Hasler, Oct 17 2022

a(n) = A007088(A033015(n)).

The number of terms with n digits is Fibonacci(n-1); the largest such term is A000042(n) = A002275(n).

A033004 Every run of digits of n in base 6 has length 2.

Original entry on oeis.org

7, 14, 21, 28, 35, 252, 266, 273, 280, 287, 504, 511, 525, 532, 539, 756, 763, 770, 784, 791, 1008, 1015, 1022, 1029, 1043, 1260, 1267, 1274, 1281, 1288, 9079, 9086, 9093, 9100, 9107, 9576, 9583, 9597, 9604, 9611, 9828, 9835

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..700

Cf. A033007, A033010.

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 6]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 7*A043310(n) (= 7*n for n<6). - M. F. Hasler, Feb 02 2014

A033005 Every run of digits of n in base 7 has length 2.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 392, 408, 416, 424, 432, 440, 784, 792, 808, 816, 824, 832, 1176, 1184, 1192, 1208, 1216, 1224, 1568, 1576, 1584, 1592, 1608, 1616, 1960, 1968, 1976, 1984, 1992, 2008, 2352, 2360, 2368, 2376, 2384, 2392

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1500

Select[Range[2500],Union[Length/@Split[IntegerDigits[#,7]]]=={2}&] (* Harvey P. Dale, Oct 24 2011 *)

a(n) = 8*A043311(n) (= 8*n for n<7). - M. F. Hasler, Feb 02 2014

A033006 Every run of digits of n in base 8 has length 2.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 576, 594, 603, 612, 621, 630, 639, 1152, 1161, 1179, 1188, 1197, 1206, 1215, 1728, 1737, 1746, 1764, 1773, 1782, 1791, 2304, 2313, 2322, 2331, 2349, 2358, 2367, 2880, 2889, 2898, 2907, 2916, 2934

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1400

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 8]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 9*A043312(n) (= 9*n for n<8). - M. F. Hasler, Feb 02 2014

Typo in name corrected by Vincenzo Librandi, Feb 05 2014

A033009 Every run of digits of n in base 11 has length 2.

Original entry on oeis.org

12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 1452, 1476, 1488, 1500, 1512, 1524, 1536, 1548, 1560, 1572, 2904, 2916, 2940, 2952, 2964, 2976, 2988, 3000, 3012, 3024, 4356, 4368, 4380, 4404, 4416, 4428, 4440, 4452, 4464, 4476

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1100

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 11]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 12*A043315(n) (= 12*n for n<11). - M. F. Hasler, Feb 02 2014

A033011 Every run of digits of n in base 13 has length 2.

Original entry on oeis.org

14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 2366, 2394, 2408, 2422, 2436, 2450, 2464, 2478, 2492, 2506, 2520, 2534, 4732, 4746, 4774, 4788, 4802, 4816, 4830, 4844, 4858, 4872, 4886, 4900, 7098, 7112, 7126, 7154

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases. See A033015 through A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1800

Select[Range[8000],Union[Length/@Split[IntegerDigits[#,13]]]=={2}&] (* Harvey P. Dale, Feb 27 2013 *)

a(n) = 14*A043317(n) (= 14*n for n<13). - M. F. Hasler, Feb 02 2014

A033012 Every run of digits of n in base 14 has length 2.

Original entry on oeis.org

15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 2940, 2970, 2985, 3000, 3015, 3030, 3045, 3060, 3075, 3090, 3105, 3120, 3135, 5880, 5895, 5925, 5940, 5955, 5970, 5985, 6000, 6015, 6030, 6045, 6060, 6075, 8820

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 04 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Select[Range[9000],Union[Length/@Split[IntegerDigits[#,14]]]=={2}&] (* Harvey P. Dale, Apr 26 2013 *)

a(n) = 15*A043318(n) (= 15*n for n<14). - M. F. Hasler, Feb 02 2014

A033013 Every run of digits of n in base 15 has length 2.

Original entry on oeis.org

16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 3600, 3632, 3648, 3664, 3680, 3696, 3712, 3728, 3744, 3760, 3776, 3792, 3808, 3824, 7200, 7216, 7248, 7264, 7280, 7296, 7312, 7328, 7344, 7360, 7376, 7392

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 04 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 15]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 16*A043319(n) (= 16n for n<15). - M. F. Hasler, Feb 02 2014

cp's OEIS Frontend

A033010 Numbers each of whose runs of digits in base 12 has length 2.

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A222813 Numbers whose binary representation is palindromic and in which all runs of 0's and 1's have length at least 2.

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A355280 Binary numbers (digits in {0, 1}) with no run of digits with length < 2.

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A033004 Every run of digits of n in base 6 has length 2.

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A033005 Every run of digits of n in base 7 has length 2.

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A033006 Every run of digits of n in base 8 has length 2.

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A033009 Every run of digits of n in base 11 has length 2.

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A033011 Every run of digits of n in base 13 has length 2.

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