A035928 -id:A035928 - cp's OEIS frontend

A044918 Positive integers whose base-2 run lengths form a palindrome.

1, 2, 3, 5, 7, 9, 10, 12, 15, 17, 21, 27, 31, 33, 38, 42, 45, 51, 52, 56, 63, 65, 73, 85, 93, 99, 107, 119, 127, 129, 142, 150, 153, 165, 170, 178, 189, 195, 204, 212, 219, 231, 232, 240, 255, 257, 273, 297, 313, 325, 341, 365, 381

Offset: 1

Clark Kimberling

This sequence exactly contains those positive integers in A006995 (positive binary palindromes) together with the terms of A035928 (those positive integers n where reversing the order of the binary digits produces the binary complement of n). - Leroy Quet, Sep 14 2009

Also the indices of the compositions that are palindromic. For the definition of the index of a composition see A298644. For example, 93 is in the sequence since its binary form is 1011101 and the composition [1,1,3,1,1] is palindromic. On the other hand, 132 is not in the sequence since its binary form is 10000100 and the composition [1,4,1,2] is not palindromic. The command c(n) from the Maple program yields the composition having index n. - Emeric Deutsch, Jan 28 2018

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic scatterplot of (n, a(n+1)-a(n)) for n = 1..9999

Cf. A006995, A035928. - Leroy Quet, Sep 14 2009

Cf. A298644, A101211. - Emeric Deutsch, Jan 28 2018

Runs:=proc(L) local j,r,i,k:j:=1: r[j]:=L[1]: for i from 2 to nops(L) do if L[i]=L[i-1] then r[j]:=r[j], L[i] else j:=j+1: r[j]:=L[i] end if end do: [seq([r[k]],k=1..j)] end proc: RunLengths:=proc(L) map(nops,Runs(L)) end  proc: c:=proc(n) ListTools:-Reverse(convert(n,base,2)): RunLengths(`%`) end proc: A:={}: for n from 1 to 500 do crev(n):=[seq(c(n)[1+ nops(c(n))-j],j=1..nops(c(n)))] od:  for n from 1 to 500 do if c(n)=crev(n) then A:=A union {n} else fi od: A; # most of the Maple program is due to W. Edwin Clark. # Emeric Deutsch, Jan 28 2018

Position[Array[Length /@ Split@ IntegerDigits[#, 2] &, 400], ? PalindromeQ, 1] // Flatten (* _Michael De Vlieger, Jan 28 2018 *)

ispal(v) = {for(i=1, #v\2, if (v[i] != v[#v-i+1], return(0));); return(1);}
isok(n) = {b = binary(n); lastb = b[1]; vrun = vector(1); vrun[1] = 1; for (i=2, #b, if (b[i] != lastb, vrun = concat(vrun, 1); lastb = b[i];, vrun[#vrun]++;)); return (ispal(vrun));} \\ Michel Marcus, Jul 10 2013

A061855 Symmetric totally balanced binary sequences: those terms of A014486 which are equal to their reversed complement.

Original entry on oeis.org

0, 2, 10, 12, 42, 52, 56, 170, 178, 204, 212, 232, 240, 682, 722, 738, 812, 852, 868, 920, 936, 976, 992, 2730, 2762, 2866, 2898, 2978, 3010, 3244, 3276, 3380, 3412, 3492, 3524, 3640, 3672, 3752, 3784, 3888, 3920, 4000, 4032, 10922, 11082, 11146

Offset: 0

Antti Karttunen, May 11 2001

nonn

These encode symmetric (palindromic) structures in many of the Catalan families, e.g. mountain ranges, parenthesizations, unlabeled rooted plane trees.

			E.g. the 45th term 11146 is 10101110001010 in binary and can be interpreted as a parenthesization: ( )( )((( )))( )( )

A. Karttunen, Some notes on Catalan's Triangle
Index entries for sequences related to parenthesizing

Obtained by "reflecting" the terms of A061854. Cf. also A035928 (ReflectBinSeq), A061856, A069766.

map(op,[seq(PalTotBalBinSequences(j),j=1..10)]);
PalTotBalBinSequences := n -> map(ReflectBinSeq,NonDivingLatticeSequences(n), n);

a(0) = 0 and the rest with the Maple function map(op, [seq(PalTotBalBinSequences(j), j=1..10)]);

A140900 A nonnegative integer n is included if the binary representation of n and the digit-reversal (with leading 0's) of the binary representation of n do not have any 1's in the same position.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 18, 24, 32, 34, 36, 38, 40, 42, 48, 52, 56, 64, 66, 68, 70, 80, 82, 96, 100, 112, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 160, 162, 168, 170, 176, 178, 192, 196, 200, 204, 208, 212, 224, 232, 240, 256, 258, 260, 262, 264

Offset: 1

Leroy Quet, Jul 24 2008

All terms of this sequence are even.
Number of terms less than or equal to 10^n: 1, 5, 26, 162, 1045, 5976, 38980, 249229, 1416583, 9381238, ..., . - Robert G. Wilson v, Aug 04 2008
Number of terms < 2^k: 1, 2, 3, 6, 18, 27, 54, 81,... (A038754). - T. D. Noe, Apr 09 2009

			36 in binary is 100100. The digit-reversal (with leading 0's) of this is 001001. These binary representations have no 1's in the same location (i.e., they can be added in binary without any carries). So 36 is in this sequence.

T. D. Noe, Table of n, a(n) for n=1..1000

A035928 is a subsequence.

fQ[n_] := Block[{id = IntegerDigits[n, 2]}, Max@ Union[id + Reverse@ id] < 2]; Select[2 Range[0, 134], fQ@# &] (* Robert G. Wilson v, Aug 04 2008 *)

is(n) = my (b=if (n, binary(n), [0])); vecmax(b+Vecrev(b))<=1 \\ Rémy Sigrist, Jun 11 2022

More terms from Robert G. Wilson v, Aug 04 2008

A195066 Numbers n such that BCR(n) is not less than n, where BCR = binary-complement-and-reverse = A036044.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 18, 20, 24, 32, 34, 36, 38, 40, 42, 44, 48, 52, 56, 64, 66, 68, 70, 72, 74, 76, 80, 82, 84, 88, 92, 96, 100, 104, 112, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 176, 178

Offset: 1

Reinhard Zumkeller, Sep 16 2011

A035928(a(n)) >= n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

a195066 n = a195066_list !! (n-1)
a195066_list = filter (\x -> a036044 x >= x) [0,2..]

A195063 Numbers n such that BCR(n) is less than n, where BCR = binary-complement-and-reverse = A036044.

Original entry on oeis.org

6, 14, 22, 26, 28, 30, 46, 50, 54, 58, 60, 62, 78, 86, 90, 94, 98, 102, 106, 108, 110, 114, 116, 118, 120, 122, 124, 126, 158, 166, 174, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 220, 222, 226, 228, 230, 234, 236, 238, 242, 244, 246, 248, 250, 252

Offset: 1

Reinhard Zumkeller, Sep 16 2011

A035928(a(n)) < n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A195066; subsequence of A195064; cf. A035928, A195065.

a195063 n = a195063_list !! (n-1)
a195063_list = filter (\x -> a036044 x < x) [0,2..]

A195064 Numbers n such that BCR(n) is not greater than n, where BCR = binary-complement-and-reverse = A036044.

Original entry on oeis.org

2, 6, 10, 12, 14, 22, 26, 28, 30, 38, 42, 46, 50, 52, 54, 56, 58, 60, 62, 78, 86, 90, 94, 98, 102, 106, 108, 110, 114, 116, 118, 120, 122, 124, 126, 142, 150, 158, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 204, 206, 210, 212, 214, 218, 220, 222, 226

Offset: 1

Reinhard Zumkeller, Sep 16 2011

A035928(a(n)) <= n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A195065; A035928 and A195063 are subsequences; cf. A195066.

a195064 n = a195064_list !! (n-1)
a195064_list = filter (\x -> a036044 x <= x) [0,2..]

A195065 Numbers n such that BCR(n) is greater than n, where BCR = binary-complement-and-reverse = A036044.

Original entry on oeis.org

0, 4, 8, 16, 18, 20, 24, 32, 34, 36, 40, 44, 48, 64, 66, 68, 70, 72, 74, 76, 80, 82, 84, 88, 92, 96, 100, 104, 112, 128, 130, 132, 134, 136, 138, 140, 144, 146, 148, 152, 154, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 208, 216, 224

Offset: 1

Reinhard Zumkeller, Sep 16 2011

A035928(a(n)) > n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A195064; subsequence of A195066.

Cf. A035928, A036044, A195063.

a195065 n = a195065_list !! (n-1)
a195065_list = filter (\x -> a036044 x > x) [0,2..]

def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)
def aupto(limit): return [m for m in range(limit+1) if BCR(m) > m]
print(aupto(224)) # Michael S. Branicky, Jun 14 2021

A277238 Folding numbers (see comments for the definition).

Original entry on oeis.org

1, 2, 6, 10, 12, 22, 28, 38, 42, 52, 56, 78, 90, 108, 120, 142, 150, 170, 178, 204, 212, 232, 240, 286, 310, 346, 370, 412, 436, 472, 496, 542, 558, 598, 614, 666, 682, 722, 738, 796, 812, 852, 868, 920, 936, 976, 992, 1086, 1134, 1206, 1254, 1338, 1386, 1458, 1506, 1596, 1644, 1716, 1764, 1848, 1896, 1968

Offset: 1

Taylor J. Smith, Oct 06 2016

Folding numbers: Numbers with an even number of bits in their binary expansion such that the XOR of the left half and the reverse of the right half is the all-1's string. Numbers with an odd number of bits in their binary expansion such that the central bit is 1, and the XOR of the left (n-1)/2 bits and the reverse of the right (n-1)/2 bits is the all-1's string.

Folding numbers with an even (resp. odd) number of bits form A035928 (resp. A276795). - N. J. A. Sloane, Nov 03 2016

			178 in base 2 is 10110010. Taking the XOR of 1011 and 0100 (which is 0010 reversed) gives the result 1111, so 178 is in the sequence.

Lars Blomberg, Table of n, a(n) for n = 1..10000
Programming Puzzles and Code Golf Stack Exchange, Folding Numbers.

Cf. A035928, A276795.

N:= 16: # to get all terms < 2^N
M[1]:= [[1]]: M[2]:= [[1,0]]:
for d from 3 to N by 2 do
  M[d]:= map(L -> [op(L[1..(d-1)/2]),1,op(L[(d+1)/2..-1])], M[d-1]);
  if d < N then
    M[d+1]:= map(L -> ([op(L[1..(d-1)/2]),0,1,op(L[(d+1)/2..-1])],[op(L[1..(d-1)/2]),1,0,op(L[(d+1)/2..-1])]), M[d-1])
  fi
od:
seq(seq(add(L[-i]*2^(i-1),i=1..d),L=M[d]),d=1..N); # Robert Israel, Nov 09 2016

{1}~Join~Select[Range@ 2000, If[OddQ@ Length@ # && Take[#, {Ceiling[ Length[#]/2]}] == {0}, False, Union[Take[#, Floor[Length[#]/2]] + Reverse@ Take[#, -Floor[Length[#]/2]]] == {1}] &@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Oct 07 2016 *)

isok(n) = {if (n==1, return(1)); b = binary(n); if ((#b % 2) && (b[#b\2+1] == 0), return (0)); vecmin(vector(#b1, k, bitxor(b[k], b[#b-k+1]))) == 1;} \\ Michel Marcus, Oct 07 2016

A276795 Folding numbers with an odd number of bits (see A277238 for definition).

Original entry on oeis.org

1, 6, 22, 28, 78, 90, 108, 120, 286, 310, 346, 370, 412, 436, 472, 496, 1086, 1134, 1206, 1254, 1338, 1386, 1458, 1506, 1596, 1644, 1716, 1764, 1848, 1896, 1968, 2016, 4222, 4318, 4462, 4558, 4726, 4822, 4966, 5062, 5242, 5338, 5482, 5578, 5746, 5842, 5986

Offset: 1

N. J. A. Sloane, Nov 03 2016

Terms greater than 1 are obtained by inserting a 1 in the middle of the binary expansions of the terms of A035928.

			78 is binary 1001110. There is a 1 in the center bit. The first 3 bits (100) and the last 3 reversed (011) sums to 111, so 78 is in the sequence.
70 is binary 1000110. There is a 0 in the center bit, thus, despite the fact that the first and last 3 bits have the same relationship as above, 70 is not in the sequence.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A035928, A053738, A277238.

{1}~Join~Select[Flatten@ Array[Range[#, 2 # - 1] &[2^#] &[2 (# - 1)] &, 7], If[OddQ@ Length@ # && Take[#, {Ceiling[Length[#]/2]}] == {0}, False, Union[Take[#, Floor[Length[#]/2]] + Reverse@ Take[#, -Floor[ Length[#]/2]]] == {1}] &@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Nov 25 2016 *)

More terms from Lars Blomberg, Nov 09 2016

A284798 Antipalindromic numbers in base 3.

Original entry on oeis.org

1, 4, 6, 13, 21, 34, 40, 46, 60, 66, 72, 97, 121, 145, 177, 201, 225, 268, 286, 304, 346, 364, 382, 424, 442, 460, 510, 528, 546, 588, 606, 624, 666, 684, 702, 781, 853, 925, 1021, 1093, 1165, 1261, 1333, 1405, 1509, 1581, 1653, 1749, 1821, 1893, 1989, 2061, 2133

Offset: 1

Paolo P. Lava, Apr 03 2017

Fixed points of the transform A284797.

A b-adic "antipalindrome" is a string of digits x where the application of the map d -> b-1-d to each digit, followed by reversal of all digits, is equal to x. This sequence lists the integers whose base-3 representation (with no leading zeros) has this property.

			34 is a term of the sequence because 34 in base 3 is 1021, its digit-by-digit complement in base 3 is 1201 and the digit reverse is again 1021.

Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv preprint arXiv:2008.06864 [math.CO], August 16 2020.

Cf. A035928, A284797.

P:=proc(q,h) local a,b,k,n; for n from 1 to q do a:=convert(n,base,h); b:=0;
for k from 1 to nops(a) do a[k]:=h-1-a[k]; b:=h*b+a[k]; od; if b=n then print(n); fi; od; end: P(10^2,8);

from itertools import count, islice
from gmpy2 import digits
def A284798_gen(): return (n for n in count(0) if not n+int((s:=digits(n,3)[::-1]),3)+1-3**len(s))
A284798_list = list(islice(A284798_gen(),35)) # Chai Wah Wu, Feb 04 2022

New name from Jeffrey Shallit, Nov 04 2023

cp's OEIS Frontend

A044918 Positive integers whose base-2 run lengths form a palindrome.

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A061855 Symmetric totally balanced binary sequences: those terms of A014486 which are equal to their reversed complement.

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A140900 A nonnegative integer n is included if the binary representation of n and the digit-reversal (with leading 0's) of the binary representation of n do not have any 1's in the same position.

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A195066 Numbers n such that BCR(n) is not less than n, where BCR = binary-complement-and-reverse = A036044.

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A195063 Numbers n such that BCR(n) is less than n, where BCR = binary-complement-and-reverse = A036044.

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A195064 Numbers n such that BCR(n) is not greater than n, where BCR = binary-complement-and-reverse = A036044.

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A195065 Numbers n such that BCR(n) is greater than n, where BCR = binary-complement-and-reverse = A036044.

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A277238 Folding numbers (see comments for the definition).

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