A048701 -id:A048701 - cp's OEIS frontend

A306607 The bottom entry in the difference table of the binary digits of n.

0, 1, 1, 0, 1, 2, -1, 0, 1, 0, 4, 3, -2, -3, 1, 0, 1, 2, -3, -2, 7, 8, 3, 4, -3, -2, -7, -6, 3, 4, -1, 0, 1, 0, 6, 5, -9, -10, -4, -5, 11, 10, 16, 15, 1, 0, 6, 5, -4, -5, 1, 0, -14, -15, -9, -10, 6, 5, 11, 10, -4, -5, 1, 0, 1, 2, -5, -4, 16, 17, 10, 11, -19

Offset: 0

Rémy Sigrist, Feb 28 2019

By convention, a(0) = 0.
For any n > 0: let (b_0, ..., b_w) be the binary representation of n:
- b_w = 1, and for any k = 0..w, 0 <= b_k <= 1,
- n = Sum_{k = 0..w} b_k * 2^k,
- a(n) is the unique value remaining after taking successively the first differences of (b_0, ..., b_w) w times.
From Robert Israel, Mar 07 2019: (Start)
  If n is odd then f(A030101(n)) = (-1)^A000523(n)*f(n).
  In particular, if n is in A048701 then a(n)=0.
  a(n) == 1 (mod A014963(A000523(n))) if n is even,
  a(n) == 0 (mod A014963(A000523(n))) if n is odd. (End)

			For n = 42:
- the binary representation of 42 is "101010",
- the corresponding difference table is:
   0   1   0   1   0   1
     1  -1   1  -1   1
      -2   2  -2   2
         4  -4   4
          -8   8
            16
- hence a(42) = 16.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000523, A007088, A030101, A030190, A030302, A048701, A014963, A187202, A241494.

f:= proc(n) local L;
  L:= convert(n,base,2);
  while nops(L) > 1 do
    L:= L[2..-1]-L[1..-2]
  od;
  op(L)
end proc:
map(f, [$0..100]); # Robert Israel, Mar 07 2019

a[n_] := NestWhile[Differences, Reverse[IntegerDigits[n, 2]], Length[#] > 1 &][[1]]; Array[a, 100, 0] (* Amiram Eldar, Mar 08 2019 *)

a(n) = if (n, my (v=Vecrev(binary(n))); while (#v>1, v=vector(#v-1, k, (v[k+1]-v[k]))); v[1], 0)

a(n) = my(b = binary(n), s = -1); sum(i = 1, #b, s=-s; binomial(#b-1, i-1) * b[i] * s) \\ David A. Corneth, Mar 07 2019

a(2^k) = 1 for any k >= 0.
a(2^k-1) = 0 for any k > 1.
a(3*2^k) = -k for any k >= 0.
a(n) = Sum_{k=0..A000523(n)} binomial(A000523(n), k)*(-1)^k*A030302(n,k). - David A. Corneth, Mar 07 2019
G.f.: 1/(x-1)*Sum_{k>=0}(x^(2^(k+1))-x^(2^k) + x^(2^k)/(x^(2^k)+1)*Sum_{m>=k+1}(binomial(m,k)*(-1)^(m-k)*(x^(2^(m+1))-x^(2^m)))). - Robert Israel, Mar 07 2019

A342036 Palindromes of even length only using 0 or 1.

Original entry on oeis.org

0, 11, 1001, 1111, 100001, 101101, 110011, 111111, 10000001, 10011001, 10100101, 10111101, 11000011, 11011011, 11100111, 11111111, 1000000001, 1000110001, 1001001001, 1001111001, 1010000101, 1010110101, 1011001101, 1011111101, 1100000011, 1100110011

Offset: 0

Seiichi Manyama, Feb 26 2021

Subsequence of A057148.

a(n) is a multiple of 11.

			A006995|A057148|A048701|A342036|A048700|A342040
-------+-------+-------+-------+-------+-------
     0 |     0 |     0 |     0 |       |
     1 |     1 |       |       |     1 |     1
     3 |    11 |     3 |    11 |       |
     5 |   101 |       |       |     5 |   101
     7 |   111 |       |       |     7 |   111
     9 |  1001 |     9 |  1001 |       |
    15 |  1111 |    15 |  1111 |       |
    17 | 10001 |       |       |    17 | 10001

Cf. A006995, A007088, A048701, A057148, A070939, A305989, A342040.

Array[FromDigits@ Join[#, Reverse[#]] &@ IntegerDigits[#, 2] &, 26, 0] (* Michael De Vlieger, Feb 26 2021 *)

def a(n): b = bin(n)[2:]; return int(b+b[::-1])
print([a(n) for n in range(27)]) # Michael S. Branicky, Feb 26 2021

def A(n)
  str = n.to_s(2)
  (str + str.reverse).to_i
end
def A342036(n)
  (0..n).map{|i| A(i)}
end
p A342036(30)

a(n) = A007088(n) * 10^A070939(n) + A305989(n).

a(n) = A007088(A048701(n)). - Michel Marcus, Feb 26 2021

Offset changed to 0 by Andrey Zabolotskiy, Dec 26 2022

A306740 Numbers k such that A306607(k) = 0.

Original entry on oeis.org

0, 3, 7, 9, 15, 31, 33, 45, 51, 63, 127, 129, 153, 165, 189, 195, 219, 231, 255, 411, 435, 511, 513, 561, 585, 633, 645, 693, 717, 765, 771, 819, 843, 891, 903, 951, 975, 1023, 2047, 2049, 2145, 2193, 2289, 2313, 2409, 2457, 2553, 2565, 2661, 2709, 2805, 2829, 2925, 2973, 3069, 3075, 3171, 3219

Offset: 1

Robert Israel, Mar 07 2019

The first even terms are 0, 68690167808, 68690561024, 68690757632, 68691150848, 68698560512, 68698953728, 68699150336, 68699543552, 68715331584, 68715724800, 68715921408, 68716314624. - Robert Israel, Mar 10 2019

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

Cf. A306607. Includes A048701.

f:= proc(n) local L;
  L:= convert(n, base, 2);
  while nops(L) > 1 do
    L:= L[2..-1]-L[1..-2]
  od;
  op(L)
end proc:
select(f=0, [$0..10000]);

seqQ[n_] := NestWhile[Differences, Reverse[IntegerDigits[n, 2]], Length[#]>1&] == {0}; Select[Range[0, 3000], seqQ] (* Amiram Eldar, Mar 08 2019 *)

A371238 Euler totient function applied to the binary palindromes of even length.

Original entry on oeis.org

2, 6, 8, 20, 24, 32, 36, 84, 96, 80, 108, 96, 144, 120, 128, 324, 320, 288, 420, 336, 360, 476, 384, 512, 432, 560, 540, 504, 632, 480, 600, 1364, 960, 1344, 1296, 1536, 1440, 1296, 1584, 1296, 1772, 1512, 1280, 1760, 1440, 1980, 1800, 1600, 1800, 2016, 1536, 1872

Offset: 1

Amiram Eldar, Mar 16 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
William D. Banks and Igor E. Shparlinski, Average value of the Euler function on binary palindromes, Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 54, No. 2 (2006), pp. 95-101; alternative link.

Cf. A000010, A006995, A048701, A059956, A070939.

EulerPhi[Select[Range[5000], EvenQ[Length[(d = IntegerDigits[#, 2])]] && PalindromeQ[d] &]]

is(n) = Vecrev(n = binary(n)) == n && !((#n)%2);
lista(kmax) = for(k = 1, kmax, if(is(k), print1(eulerphi(k), ", ")));

a(n) = A000010(A048701(n)).

(1/N(k)) * Sum_{j, A070939(A048701(j)) = 2*k} a(j) = 3 * 2^(2*k-2) * (6/Pi^2 + O((k/log(k))^(-1/4))), where N(k) = Sum_{j, A070939(A048701(j)) = 2*k} 1 (Banks and Shparlinski, 2006).

cp's OEIS Frontend

A306607 The bottom entry in the difference table of the binary digits of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A342036 Palindromes of even length only using 0 or 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Ruby

Formula

Extensions

A306740 Numbers k such that A306607(k) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A371238 Euler totient function applied to the binary palindromes of even length.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula