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 A281392 Number of occurrences of "01" as a subsequence in the binary expansion of n. 1
 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 4, 3, 4, 1, 5, 4, 7, 3, 6, 5, 7, 2, 5, 4, 6, 3, 5, 4, 5, 1, 6, 5, 9, 4, 8, 7, 10, 3, 7, 6, 9, 5, 8, 7, 9, 2, 6, 5, 8, 4, 7, 6, 8, 3, 6, 5, 7, 4, 6, 5, 6, 1, 7, 6, 11, 5, 10, 9, 13, 4, 9, 8, 12, 7, 11, 10, 13, 3, 8, 7, 11, 6, 10, 9, 12, 5, 9, 8, 11, 7, 10, 9, 11, 2, 7, 6, 10, 5, 9, 8, 11, 4, 8, 7, 10, 6, 9, 8, 10, 3, 7, 6, 9, 5, 8, 7, 9, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS By "subsequence" we do not demand that occurrences are contiguous. Furthermore, we assume that the binary expansion of n begins with a 0, and is read starting with the most significant digit. LINKS Robert Israel, Table of n, a(n) for n = 0..10000 FORMULA Completely defined by the recurrence relations a(2n) = a(n); a(4n+3) = -a(n) + 2a(2n+1); a(8n+1) = -a(2n+1) + 2a(4n+1); a(16n+5) = -2a(2n+1) + 2a(4n+1) + a(8n+5); a(16n+13) = -a(n) + a(2n+1) + a(8n+5) with a(0) = 0, a(1) = 1 and a(5) = 3. EXAMPLE For n = 5, the number of occurrences of 01 as a subsequence of 0101 is 3. MAPLE f:= proc(n) option remember; if n::even then procname(n/2) elif n mod 4 = 3 then - procname((n-3)/4) + 2*procname((n-1)/2) elif n mod 8 = 1 then -procname((n+3)/4) + 2*procname((n+1)/2) elif n mod 16 = 5 then -2*procname((n+3)/8) + 2*procname((n-1)/4) + procname((n+5)/2) elif n mod 16 = 13 then -procname((n-13)/16) +procname((n-5)/8) + procname((n-3)/2) fi; end proc: f(0):= 0: f(1):= 1: f(5):= 3: map(f, [\$0..200]); # Robert Israel, Mar 11 2020 MATHEMATICA f[n_] := f[n] = Which[ EvenQ[n], f[n/2], Mod[n, 4] == 3, -f[(n-3)/4] + 2*f[(n-1)/2], Mod[n, 8] == 1, -f[(n+3)/4] + 2*f[(n+1)/2], Mod[n, 16] == 5, -2*f[(n+3)/8] + 2*f[(n-1)/4] + f[(n+5)/2], Mod[n, 16] == 13, -f[(n-13)/16] + f[(n-5)/8] + f[(n-3)/2]]; f = 0; f = 1; f = 3; Map[f, Range[0, 200]] (* Jean-François Alcover, Sep 19 2022, after Robert Israel *) CROSSREFS Cf. A055941, which gives the analogous sequence for the pattern "10" instead of "01". Cf. A000079 (a(n)=1), A007283 (a(n)=2), A070875 (a(n)=3). Sequence in context: A070940 A020651 A294442 * A287051 A002487 A318509 Adjacent sequences: A281389 A281390 A281391 * A281393 A281394 A281395 KEYWORD nonn AUTHOR Jeffrey Shallit, Jan 21 2017 STATUS approved

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Last modified October 4 01:05 EDT 2023. Contains 365872 sequences. (Running on oeis4.)