A033001 -id:A033001 - cp's OEIS frontend

A043307 a(n) = A033001(n)/4.

1, 2, 9, 11, 18, 19, 82, 83, 99, 100, 163, 164, 171, 173, 738, 740, 747, 748, 892, 893, 900, 902, 1467, 1469, 1476, 1477, 1540, 1541, 1557, 1558, 6643, 6644, 6660, 6661, 6724, 6725, 6732, 6734, 8028, 8030, 8037, 8038, 8101, 8102, 8118, 8119

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 9, have only digits 0, 1 or 2, and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A043308 - A043320, A043291, A033001 - A033014, A033016 - A033029.

A[1]:= [1,2]:
for d from 2 to 6 do
  A[d]:= map(t -> seq(9*t+j,j=subs(t mod 9 = NULL, [0,1,2])), A[d-1])
od:
seq(op(A[d]),d=1..6); # Robert Israel, Jan 29 2017

Table[FromDigits[#,9]&/@Select[Tuples[{0,1,2},n],Min[Abs[Differences[#]]]>0&],{n,2,5}]// Flatten// Union (* Harvey P. Dale, May 27 2023 *)

is_A043307(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],9))[2]<3 && n[1]%3!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

a(n) = my(v=binary(n+1)); v[1]=0; for(i=2,#v, v[i]+=(v[i]>=v[i-1])); fromdigits(v,9); \\ Kevin Ryde, Mar 13 2021

From Robert Israel, Jan 29 2017: (Start)
If a(n) == 0 (mod 3) then a(2*n+1) = 9*a(n) + 1 else a(2*n+1) = 9*a(n).
If a(n) == 2 (mod 3) then a(2*n+2) = 9*a(n) + 1 else a(2*n+1) = 9*a(n)+2.
a(4k+5) = 9*a(2k+2).
(End)

A043291 Every run length in base 2 is 2.

Original entry on oeis.org

3, 12, 51, 204, 819, 3276, 13107, 52428, 209715, 838860, 3355443, 13421772, 53687091, 214748364, 858993459, 3435973836, 13743895347, 54975581388, 219902325555, 879609302220, 3518437208883, 14073748835532, 56294995342131, 225179981368524, 900719925474099

Offset: 1

Clark Kimberling

a(n) is the number whose binary representation is A153435(n). - Omar E. Pol, Jan 18 2009

See A033001 and following for the analog in other bases and the variant with runs of length >= 2. - M. F. Hasler, Feb 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Cf. A153435 (binary).
Bisections: A108020, A182512. Bisection of A077854.
Cf. A000975, A033001, A033114.

[Floor(4^(n+1)/5): n in [1..30]]; // Vincenzo Librandi, Jun 26 2011

seq(floor(4^(n+1)/5),n=1..25); # Mircea Merca, Dec 26 2010

f[n_] := Floor[4^(n + 1)/5]; Array[f, 23] (* or *)
a[1] = 3; a[2] = 12; a[3] = 51; a[n_] := a[n] = 4 a[n - 1] + a[n - 2] - 4 a[n - 3]; Array[a, 23] (* or *)
3 LinearRecurrence[{4, 1, -4}, {1, 4, 17}, 23] (* Robert G. Wilson v, Jul 01 2014 *)

A043291 = n->4^(n+1)\5 \\ M. F. Hasler, Feb 01 2014

def a(n): return int(''.join([['11', '00'][i%2] for i in range(n)]), 2)
print([a(n) for n in range(1, 26)]) # Michael S. Branicky, Mar 12 2021

a(n) = 4*a(n-1)+a(n-2)-4*a(n-3), n>3. - John W. Layman, Feb 01 2000
a(n) = floor(4^(n+1)/5). - Mircea Merca, Dec 26 2010
G.f.: 3*x / ( (x-1)*(4*x-1)*(1+x) ). - Joerg Arndt, Jan 08 2011
a(n) = 3*A033114(n). - R. J. Mathar, Jan 08 2011

A033014 Every run of digits of n in base 16 has length 2.

Original entry on oeis.org

17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 4352, 4386, 4403, 4420, 4437, 4454, 4471, 4488, 4505, 4522, 4539, 4556, 4573, 4590, 4607, 8704, 8721, 8755, 8772, 8789, 8806, 8823, 8840, 8857, 8874

Offset: 1

Clark Kimberling

			In base 16, a(1)=17 is written 11; the subsequent 14 values are the multiples of 17, corresponding to 22, 33, 44, ..., FF.
This is followed by a(16) = 4352 = 1100[16], then (still in base 16): 1122, 1133,..., 11FF, 2200, 2211, 2233, etc...

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

See A033001 for further cross-references.

Select[Range[9000],Union[Length/@Split[IntegerDigits[#,16]]]=={2}&] (* Harvey P. Dale, Jan 19 2013 *)

from itertools import count, islice, groupby
def A033014_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:set(len(list(g)) for k, g in groupby(hex(n)[2:]))=={2},count(max(startvalue,1)))
A033014_list = list(islice(A033014_gen(),20)) # Chai Wah Wu, Mar 10 2023

a(n) = 17*A043320(n) (= 17n for n<16, cf Example). - M. F. Hasler, Feb 02 2014

A033015 Numbers whose base-2 expansion has no run of digits with length < 2.

Original entry on oeis.org

3, 7, 12, 15, 24, 28, 31, 48, 51, 56, 60, 63, 96, 99, 103, 112, 115, 120, 124, 127, 192, 195, 199, 204, 207, 224, 227, 231, 240, 243, 248, 252, 255, 384, 387, 391, 396, 399, 408, 412, 415, 448, 451, 455, 460, 463, 480, 483, 487, 496, 499, 504, 508, 511, 768

Offset: 1

Clark Kimberling

See A033016 and following for the variants in other bases, A043291 for run lengths equal to 2 (which has a very simple formula) and A033001 and following for the analog of the latter in other bases. - M. F. Hasler, Feb 01 2014
The number zero also satisfies the definition if we consider that its base-2 expansion is empty. - M. F. Hasler, Oct 06 2022
If we define row n as subset of terms with n bits, i.e., 2^(n-1) < a(k) < 2^n, then we get row n by duplicating the last bit (LSB) of the terms in row n-1 and appending twice the negated LSB to the terms in row n-2. This gives the FORMULA for the number of terms in row n. - M. F. Hasler, Oct 17 2022

			The first terms, written in binary, are: 11, 111, 1100, 1111, 11000, 11100, 11111, 110000, 110011, ...; cf. sequence A355280. - _M. F. Hasler_, Oct 06 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A355280 (in binary).
Cf. A222813 (palindromes subsequence).
Cf. A033016, A043291.
See A033001 for further cross-references.

Select[Range[2000], Min[Length/@Split[IntegerDigits[#, 2]]]>1&] (* Vincenzo Librandi, Feb 05 2014 *)

is(n)=my(t); if(n%2, t=valuation(n+1,2); if(t==1,return(0)); n>>=t); while(n, t=valuation(n,2); if(t==1,return(0)); n>>=t; t=valuation(n+1,2); if(t==1,return(0)); n>>=t); 1 \\ Charles R Greathouse IV, Mar 29 2013

select( is_A033015(n)=!bitand(n=bitxor(n,n<<1),n<<1)&&bitand(n,3)!=2, [1..770]) \\ M. F. Hasler, Oct 06 2022 (replacing less efficient code from 2014)

{A033015_row(n)=if(n>3, setunion([x*2+x%2|x<-A033015_row(n-1)], [x*4+3-x%2*3|x<-A033015_row(n-2)]), n>1, [2^n-1], [])} \\ "Row" of n-digit terms. For (very) large n one could use memoization rather than this naive recursive definition.
concat(apply(A033015_row, [1..9])) \\ To get the "flattened" sequence. - M. F. Hasler, Oct 17 2022

from itertools import groupby
def ok(n): return all(len(list(g)) >= 2 for k, g in groupby(bin(n)[2:]))
print([i for i in range(1, 769) if ok(i)]) # Michael S. Branicky, Jan 04 2021

def A033015_row(n): # terms with n bits <=> in [2^(n-1) .. 2^n]
    return [[], [], [3], [7]][n] if n < 4 else sorted(
    [x*2+x%2 for x in A033015_row(n-1)] +
    [x*4+3-x%2*3 for x in A033015_row(n-2)]) # M. F. Hasler, Oct 17 2022
print(sum((A033015_row(n)for n in range(11)),[]))

The number of n-bit terms is Fibonacci(n-1) = A000045(n-1). - M. F. Hasler, Oct 17 2022

Extended by Ray Chandler, Dec 18 2009

A033016 Numbers whose base-3 expansion has no run of digits with length < 2.

Original entry on oeis.org

4, 8, 13, 26, 36, 40, 44, 72, 76, 80, 108, 117, 121, 125, 134, 216, 229, 234, 238, 242, 324, 328, 332, 351, 360, 364, 368, 377, 396, 400, 404, 648, 652, 656, 684, 688, 692, 702, 715, 720, 724, 728, 972, 976, 980, 985, 998, 1053

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007089. Supersequence of A033001.

Select[Range[10000], Min[Length/@Split[IntegerDigits[#, 3]]]>1&] (* Vincenzo Librandi, Feb 05 2014 *)

A043320 Numbers which, written in base 256, have all digits less than 16 and no two adjacent digits equal.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 256, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 512, 513, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 768, 769, 770, 772, 773, 774

Offset: 1

Clark Kimberling

Sequence A033014 consists of the numbers that have all base 16 digits repeated *exactly* twice. (This is equivalent to say that the base-256 digits are 0x00, 0x11, 0x22,... or 0xFF, in hex notation, and no two adjacent base-256 digits are equal.) Thus, these numbers are divisible by 0x11 = 17, and the result of the division is a number which has no other base-256 digits than 0x00, 0x01,... or 0x0F, and no two adjacent digits equal. Conversely, it is clear that exactly these numbers are terms of A033014 when multiplied by 17 = 0x11. - M. F. Hasler, Feb 05 2014

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

Cf. A043307 - A043319, A043291, A033001 - A033014, A033015 - A033029.

Select[Range[20000], Union[Length/@Split[IntegerDigits[#, 16]]]=={2}&]/17 (* Vincenzo Librandi, Feb 06 2014 *)

is_A043320(n)={(n=[n])&&!until(!n[1], ((n=divrem(n[1], 256))[2]<16 && n[1]%16!=n[2])||return)} \\ M. F. Hasler, Feb 03 2014

from itertools import count, islice, groupby
def A043320_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:set(len(list(g)) for k, g in groupby(hex(17*n)[2:]))=={2},count(max(startvalue,1)))
A043320_list = list(islice(A043320_gen(),20)) # Chai Wah Wu, Mar 10 2023

a(n) = A033014(n)/17. [This was initially the definition of the sequence. - M. F. Hasler, Feb 03 2014]

New definition by M. F. Hasler, Feb 03 2014

A033002 Every run of digits of n in base 4 has length 2.

Original entry on oeis.org

5, 10, 15, 80, 90, 95, 160, 165, 175, 240, 245, 250, 1285, 1290, 1295, 1440, 1445, 1455, 1520, 1525, 1530, 2565, 2570, 2575, 2640, 2650, 2655, 2800, 2805, 2810, 3845, 3850, 3855, 3920, 3930, 3935, 4000, 4005, 4015, 20560

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..600

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 4]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 5*A043308(n) (= 5*n for n<4). - M. F. Hasler, Feb 04 2014

A033008 Every run of digits of n in base 10 has length 2.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1100, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 2200, 2211, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 3300, 3311, 3322, 3344, 3355, 3366, 3377, 3388, 3399, 4400, 4411, 4422, 4433

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..800

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 10]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 11*A043314(n) (= 11*n for n<10). - M. F. Hasler, Feb 02 2014

A033003 Every run of digits of n in base 5 has length 2.

Original entry on oeis.org

6, 12, 18, 24, 150, 162, 168, 174, 300, 306, 318, 324, 450, 456, 462, 474, 600, 606, 612, 618, 3756, 3762, 3768, 3774, 4050, 4056, 4068, 4074, 4200, 4206, 4212, 4224, 4350, 4356, 4362, 4368, 7506, 7512, 7518, 7524, 7650, 7662

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..1000

Select[Range[10000],Union[Length/@Split[IntegerDigits[#, 5]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 6*A043309(n) (= 6*n for n<5). - M. F. Hasler, Feb 02 2014

A033007 Every run of digits of n in base 9 has length 2.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 810, 830, 840, 850, 860, 870, 880, 890, 1620, 1630, 1650, 1660, 1670, 1680, 1690, 1700, 2430, 2440, 2450, 2470, 2480, 2490, 2500, 2510, 3240, 3250, 3260, 3270, 3290, 3300, 3310, 3320, 4050

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..1000

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 9]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 10*A043313(n) (= 10*n for n<9). - M. F. Hasler, Feb 02 2014

cp's OEIS Frontend

A043307 a(n) = A033001(n)/4.

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Maple

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A043291 Every run length in base 2 is 2.

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Magma

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A033014 Every run of digits of n in base 16 has length 2.

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A033015 Numbers whose base-2 expansion has no run of digits with length < 2.

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A033016 Numbers whose base-3 expansion has no run of digits with length < 2.

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Mathematica

A043320 Numbers which, written in base 256, have all digits less than 16 and no two adjacent digits equal.

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A033002 Every run of digits of n in base 4 has length 2.

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