A206920 -id:A206920 - cp's OEIS frontend

A206913 Greatest binary palindrome <= n; the binary palindrome floor function.

0, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 9, 9, 9, 9, 15, 15, 17, 17, 17, 17, 21, 21, 21, 21, 21, 21, 27, 27, 27, 27, 31, 31, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 45, 45, 45, 45, 45, 45, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 63, 63, 65, 65, 65, 65

Offset: 0

Hieronymus Fischer, Feb 13 2012

Also the greatest binary palindrome < n + 1;

For n > 0, a(n-1) is the greatest binary palindrome < n.

			a(0) = 0 since 0 is the greatest binary palindrome <= 0;
a(1) = 1 since 1 is the greatest binary palindrome <= 1;
a(2) = 1 since 1 is the greatest binary palindrome <= 2;
a(3) = 3 since 3 is the greatest binary palindrome <= 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A006995, A206915, A206920, A030301.

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

a206913 n = last $ takeWhile (<= n) a006995_list
-- Reinhard Zumkeller, Feb 27 2012

Let n > 2, p = 1 + 2*floor((n-1)/2), m = floor(log_2(p)), q = floor((m+1)/2), s = floor(log_2(p-2^q)),
F(x, r) = floor(x/2^q)*2^q + Sum_{k = 0...q - 1} (floor(x/2^(r-k)) mod 2)*2^k;
If F(p, m) <= n then a(n) = F(p, m), otherwise a(n) = F(p-2^q, s).
By definition: F(p, m) = floor(p/2^q)*2^q + A030101(p) mod 2^q; also: F(p-2^q, s) = floor((p-2^q)/2^q)*2^q + A030101(p-2^q) mod 2^q; [Edited and corrected by Hieronymus Fischer, Sep 08 2018]
a(n) = A006995(A206915(n));
a(n) = A006995(A206915(A206914(n+1))-1);
a(n) = A006995(A206916(A206914(n+1))-1).

A206914 Least binary palindrome >= n; the binary palindrome ceiling function.

Original entry on oeis.org

0, 1, 3, 3, 5, 5, 7, 7, 9, 9, 15, 15, 15, 15, 15, 15, 17, 17, 21, 21, 21, 21, 27, 27, 27, 27, 27, 27, 31, 31, 31, 31, 33, 33, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 51, 51, 51, 51, 51, 51, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 65, 65, 73, 73

Offset: 0

Hieronymus Fischer, Feb 15 2012

For n > 0 also the least binary palindrome > n - 1;

a(n+1) is the least binary palindrome > n

			a(0) = 0 since 0 is the least binary palindrome >= 0;
a(1) = 1 since 1 is the least binary palindrome >= 1;
a(2) = 3 since 3 is the least binary palindrome >= 2;
a(5) = 5 since 5 is the least binary palindrome >= 5;

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A206915, A206920, A006995.

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

a206914 n = head $ dropWhile (< n) a006995_list
-- Reinhard Zumkeller, Feb 27 2012

a(n) = A006995(A206916(n));
a(n) = A006995(A206916(A206913(n-1))+1);
a(n) = A006995(A206915(A206913(n-1))+1);

A206915 The index (in A006995) of the greatest binary palindrome <= n; also the 'lower inverse' of A006995.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Hieronymus Fischer, Feb 15 2012

The greatest m such that A006995(m)<= n;
The number of binary palindromes <= n;
n is palindromic iff a(n)=A206916(n);
a(n) is the number of the binary palindrome A206913(n);
if n is a binary palindrome, then A006995(a(n))=n, so a(n) is 'inverse' with respect to A006995.
Partial sums of the binary palindromic characteristic function A178225.

			a(1)=2 since 2 is the index number of the greatest binary palindrome <= 1;
a(5)=4 since there are only 4 binary palindromes (namely 0,1,3 and 5) which are less than or equal to 5;
a(10)=6 since A006995(6)=9<=10, but A006995(7)=15>10, and so that, 6 is the index number of greatest binary palindrome <= 10;

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A006995, A206914, A206916, A206920.

A178225[n_]:=Boole[PalindromeQ[IntegerDigits[n,2]]];
Accumulate[Array[A178225,100,0]] (* Paolo Xausa, Oct 15 2023 *)

def A206915(n):
    l = n.bit_length()
    k = l+1>>1
    return (n>>l-k)-(int(bin(n)[k+1:1:-1] or '0',2)>(n&(1<Chai Wah Wu, Jul 24 2024

a(n) = max(m | A006995(m) <= n);
a(A006995(n)) = n;
A006995(a(n)) <= n, equality holds true iff n is a binary palindrome;
Let p = A206913(n), m = floor(log_2(p)) and p>2, then:
a(n) = (((5-(-1)^m)/2) + sum_{k=1..floor(m/2)} (floor(p/2^k) mod 2)/2^k)) * 2^floor(m/2).
a(n) = (1/2)*((6-(-1)^m)*2^floor(m/2) - 1 - sum_{k=1..floor(m/2)} (-1)^floor(p/2^k) * 2^(floor(m/2)-k))).
a(n) = (5-(-1)^m) * 2^floor(m/2)/2 - 3*sum_{k=2..floor(m/2)} (floor(p/2^k) * 2^floor(m/2)/2^k) + (floor(p/2) * 2^floor(m/2)/2 - 2*floor((p/2) * 2^floor(m/2)) * floor((m-1)/m+1/2).
Partial sums S(n) = sum_{k=0..n} a(k):
S(n) = (n+1)*a(n) - A206920(a(n)).
G.f.: g(x) = (1+x+x^3+sum_{j>=1} x^(3*2^j)*(f_j(x)+f_j(1/x)))/(1-x), where the f_j(x) are defined as follows:
  f_1(x) = x, and for j>1,
  f_j(x) = x^3*product_{k=1..floor((j-1)/2)} (1+x^b(j,k)), where b(j,k)=2^(floor((j-1)/2)-k)*((3+(-1)^j)*2^(2*k+1)+4) for k>1, and b(j,1)=(2+(-1)^j)*2^(floor((j-1)/2)+1).

A206916 Index of the least binary palindrome >=n; also the "upper inverse" of A006995.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 17, 17, 17, 17, 17

Offset: 0

Hieronymus Fischer, Feb 17 2012

The least m such that A006995(m)>=n;
n is palindromic iff a(n)=A206915(n);
a(n) is the number of the binary palindrome A206914(n);
if n is a binary palindrome, then A006995(a(n))=n, so a(n) is 'inverse' with respect to A006995

			a(2)=3 since 3 is the index number of the least binary palindrome >= 2;
a(5)=4 since 4 is the index number of the least binary palindrome >= 5;
a(10)=7 since A006995(7)=15>=10, but A006995(6)=9<10, and so that, 7 is the index number of least binary palindrome >= 10;

Cf. A006995, A206920.

def A206916(n):
    l = n.bit_length()
    k = l+1>>1
    return (n>>l-k)+(int(bin(n)[k+1:1:-1] or '0',2)<(n&(1<Chai Wah Wu, Jul 24 2024

a(n)=min(m|A006995(m)>=n);
a(A006995(n))=n;
A006995(a(n))>=n, equality holds true iff n is a binary palindrome;
Let p=A206914(n), m=floor(log_2(p)) and p>2, then:
a(n)=(((5-(-1)^m)/2) + sum_{k=1..floor(m/2)} (floor(p/2^k) mod 2)/2^k))*2^floor(m/2);
a(n)=(1/2)*((6-(-1)^m)*2^floor(m/2)-1-sum_       {k=1..floor(m/2)} (-1)^floor(p/2^k)*2^(floor(m/2)-k)));
a(n)=(5-(-1)^m)*2^floor(m/2)/2-3*sum_{k=2..floor(m/2)} floor(p/2^k)*2^floor(m/2)/2^k)+(floor(p/2)*2^floor(m/2)/2-2*floor((p/2)*2^floor(m/2))*floor((m-1)/m+1/2).
Partial sums S(n) = sum_{k=0..n} a(k):
S(n) = 1+n*a(n)-A206920(a(n)-1), valid for n>0.
G.f.: g(x)=(x+x^2+x^3+sum_{j=1..infinity} x^(3*2^j)*(f_j(x)+f_j(1/x)))/(x(1-x)), where the f_j(x) are defined as follows:
  f_1(x)=x, and for j>1,
  f_j(x)=x^3*product_{k=1..floor((j-1)/2)} (1+x^b(j,k)), where b(j,k)=2^(floor((j-1)/2)-k)*((3+(-1)^j)*2^(2*k+1)+4) for k>1, and b(j,1)=(2+(-1)^j)*2^(floor((j-1)/2)+1).

A206919 Sum of binary palindromes <= n.

Original entry on oeis.org

0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25, 25, 25, 25, 25, 40, 40, 57, 57, 57, 57, 78, 78, 78, 78, 78, 78, 105, 105, 105, 105, 136, 136, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 214, 214, 214, 214, 214, 214, 265, 265, 265, 265, 265, 265, 265, 265

Offset: 0

Hieronymus Fischer, Feb 18 2012

Sum of binary palindromes A006995(k) <= n.

Different from A206920.

			a(2)=1, since the only binary palindromes <= 1 are p=0 and p=1;
a(5)=9, since the sum of all binary palindromes <= 5 is 9 = 0 + 1 + 3 + 5.

Cf. A006995, A206913, A206915.

a(n) = sum(k=1, n, my(b=binary(k)); if (b==Vecrev(b), k)); \\ Michel Marcus, Sep 09 2018

a(n) = Sum_{k=1..A206915(A206913(n))} A006995(k).
a(n) = A206920(A206915(A206913(n))).
Let p = A206913(n) > 3, m = floor(log_2(p)), then
a(n) = (8/7)*((3/4)*(4-(-1)^m)/(3+(-1)^m)*2^(3*floor(m/2))-1) + (floor(p/2^floor(m/2)) mod 2)*p + 2^m + 1 + Sum_{k=1..floor(m/2)-1} (floor(p/2^k) mod 2)*(2^k+2^(m-k)+2^(m-floor(m/2)+1)*(4^(floor(m/2)-k-1)-1)+(2-(-1)^m)*2^floor(m/2)+2^(floor(m/2)-k)*(p-floor((p mod (2^(m-k+1)))/2^k)*2^k)). - [Corrected; missing factor to the sum term (2-(-1)^m) pasted by the author, Sep 08 2018]

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A206913 Greatest binary palindrome <= n; the binary palindrome floor function.

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A206914 Least binary palindrome >= n; the binary palindrome ceiling function.

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A206915 The index (in A006995) of the greatest binary palindrome <= n; also the 'lower inverse' of A006995.

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A206916 Index of the least binary palindrome >=n; also the "upper inverse" of A006995.

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A206919 Sum of binary palindromes <= n.

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