The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A206915 The index (in A006995) of the greatest binary palindrome <= n; also the 'lower inverse' of A006995. 13
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. LINKS Paolo Xausa, Table of n, a(n) for n = 0..10000 FORMULA 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). EXAMPLE 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; MATHEMATICA A178225[n_]:=Boole[PalindromeQ[IntegerDigits[n, 2]]]; Accumulate[Array[A178225, 100, 0]] (* Paolo Xausa, Oct 15 2023 *) PROG (Python) 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<

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 3 23:57 EDT 2024. Contains 374905 sequences. (Running on oeis4.)