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 A206915 The index (in A006995) of the greatest binary palindrome <= n; also the 'lower inverse' of A006995. 13
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 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 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; CROSSREFS Cf. A006995, A206914, A206915, A206916, A206920. Sequence in context: A194964 A029923 A096029 * A327247 A210609 A210608 Adjacent sequences:  A206912 A206913 A206914 * A206916 A206917 A206918 KEYWORD nonn,base AUTHOR Hieronymus Fischer, Feb 15 2012 STATUS approved

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Last modified July 12 23:28 EDT 2020. Contains 335669 sequences. (Running on oeis4.)