A033016 -id:A033016 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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)

A048328 Numbers that are repdigits in base 3.

Original entry on oeis.org

0, 1, 2, 4, 8, 13, 26, 40, 80, 121, 242, 364, 728, 1093, 2186, 3280, 6560, 9841, 19682, 29524, 59048, 88573, 177146, 265720, 531440, 797161, 1594322, 2391484, 4782968, 7174453, 14348906, 21523360, 43046720, 64570081, 129140162, 193710244, 387420488, 581130733

Offset: 0

Patrick De Geest, Feb 15 1999

Case for base 2 see A000225: 2^n - 1.

If the sequence b(n) represents the number of paths of length n, n >= 1, starting at node 1 and ending at nodes 1, 2, 3 and 4 on the path graph P_5 then a(n-1) = b(n) - 1. - Johannes W. Meijer, May 29 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Repdigit.
Index entries for 3-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

Bisections: A024023 and A003462.

Cf. A000225, A010785, A028987, A028988, A033016, A038754, A068911, A124302, A214369.

nmax := 35; a(0) := 0: for n from 1 to nmax do a(2*n) := a(2*n-2) + 2*3^(n-1); od: a(1) := 1: for n from 1 to nmax do a(2*n+1) := 1*a(2*n-1) + 3^n; od: seq(a(n), n=0..nmax);
# End program 1
with(GraphTheory): G := PathGraph(5): A:= AdjacencyMatrix(G): nmax := nmax; for n from 1 to nmax+1 do B(n) := A^n; b(n) := add(B(n)[1, k], k=1..4); a1(n-1) := b(n)-1; od: seq(a1(n), n=0..nmax);
# End program 2
# From Johannes W. Meijer, May 29 2010, revised Sep 23 2012
# third Maple program:
a:= n->(<<0|1>, <-3|4>>^iquo(n, 2, 'r').`if`(r=0, <<0, 2>>, <<1, 4>>))[1, 1]:
seq (a(n), n=0..60);  # Alois P. Heinz, Sep 23 2012

Rest[FromDigits[#, 3]&/@Flatten[Table[{PadRight[{1}, n, 1], PadRight[{2}, n, 2]}, {n, 0, 20}], 1]] (* Harvey P. Dale, Feb 03 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -3,0,4,0]^n*[0;1;2;4])[1,1] \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (2*x^2+x)/(1-4*x^2+3*x^4). - Alois P. Heinz, Sep 23 2012
Sum_{n>=1} 1/a(n) = 3 * A214369 = 2.04646050781571420028... - Amiram Eldar, Jan 21 2022
a(n) = (3^(n/2)*(sqrt(3) + 2 - (-1)^n*(sqrt(3) - 2)) - 3 - (-1)^n)/4. - Stefano Spezia, Feb 18 2022

A043308 a(n)=A033002(n)/5.

Original entry on oeis.org

1, 2, 3, 16, 18, 19, 32, 33, 35, 48, 49, 50, 257, 258, 259, 288, 289, 291, 304, 305, 306, 513, 514, 515, 528, 530, 531, 560, 561, 562, 769, 770, 771, 784, 786, 787, 800, 801, 803, 4112, 4114, 4115, 4128, 4129, 4131, 4144, 4145

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 16, have all digits less than 4 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

is_A043308(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],16))[2]<4 && n[1]%4!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

A043312 a(n) = A033006(n)/9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 64, 66, 67, 68, 69, 70, 71, 128, 129, 131, 132, 133, 134, 135, 192, 193, 194, 196, 197, 198, 199, 256, 257, 258, 259, 261, 262, 263, 320, 321, 322, 323, 324, 326, 327, 384, 385, 386, 387, 388, 389, 391, 448, 449

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 64, have only digits 0 through 7, and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

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

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

f:= proc(n) local i;
      seq(64*n+i, i= subs(n mod 64 = NULL, [$0..7]))
end proc:
A:= $1..7: R:= [A]:
for d from 2 to 3 do
  R:= map(f, R);
  A:= A, op(R);
od:
A; # Robert Israel, Jun 11 2019

is_A043312(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],64))[2]<8 && n[1]%8!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

A043317 a(n)=A033011(n)/14.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 169, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 338, 339, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 507, 508, 509, 511, 512, 513, 514, 515, 516, 517, 518, 519, 676, 677, 678

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 169, have all digits less than 13 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

Select[Range[700],Max[IntegerDigits[#,169]]<13&&SequenceCount[ IntegerDigits[ #,169],{x_,x_}]==0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 07 2018 *)

is_A043317(n)=(n=[n])&&!until(!n[1],((n=divrem(n,169))[2]<13 && n[2]!=n[1]%13)||return) \\ M. F. Hasler, Feb 03 2014

A043319 a(n)=A033013(n)/16.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 225, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 450, 451, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 675, 676, 677, 679, 680, 681, 682, 683, 684

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 225, have all digits less than 15 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

is_A043319(n)=(n=[n])&&!until(!n[1], ((n=divrem(n[1], 225))[2]<15 && n[1]%15!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

A043309 a(n)=A033003(n)/6.

Original entry on oeis.org

1, 2, 3, 4, 25, 27, 28, 29, 50, 51, 53, 54, 75, 76, 77, 79, 100, 101, 102, 103, 626, 627, 628, 629, 675, 676, 678, 679, 700, 701, 702, 704, 725, 726, 727, 728, 1251, 1252, 1253, 1254, 1275, 1277, 1278, 1279, 1325, 1326, 1327, 1329

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 25, have all digits less than 5 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

is_A043309(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],25))[2]<5 && n[1]%5!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

A043310 a(n)=A033004(n)/7.

Original entry on oeis.org

1, 2, 3, 4, 5, 36, 38, 39, 40, 41, 72, 73, 75, 76, 77, 108, 109, 110, 112, 113, 144, 145, 146, 147, 149, 180, 181, 182, 183, 184, 1297, 1298, 1299, 1300, 1301, 1368, 1369, 1371, 1372, 1373, 1404, 1405, 1406, 1408, 1409, 1440, 1441, 1442, 1443, 1445, 1476, 1477

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 36, have all digits less than 6 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

is_A043310(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],36))[2]<6 && n[1]%6!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

Definition corrected by and more terms from Georg Fischer, Mar 04 2021

A043311 a(n)=A033005(n)/8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 49, 51, 52, 53, 54, 55, 98, 99, 101, 102, 103, 104, 147, 148, 149, 151, 152, 153, 196, 197, 198, 199, 201, 202, 245, 246, 247, 248, 249, 251, 294, 295, 296, 297, 298, 299, 2402, 2403, 2404, 2405, 2406, 2407, 2499

Offset: 1

Clark Kimberling

Numbers which, written in base 49, have only digits 0 through 6 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029. ~

is_A043311(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],49))[2]<7 && n[1]%7!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Python

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A048328 Numbers that are repdigits in base 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A043308 a(n)=A033002(n)/5.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A043312 a(n) = A033006(n)/9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

A043317 a(n)=A033011(n)/14.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A043319 a(n)=A033013(n)/16.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A043309 a(n)=A033003(n)/6.