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A014082 Number of occurrences of '111' in binary expansion of n.

%I A014082 #50 Feb 16 2025 08:32:32
%S A014082 0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,1,0,0,0,0,1,1,2,3,0,0,
%T A014082 0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,4,0,0,0,0,
%U A014082 0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,1,0,0,0,0,1,1,2,3,0,0,0,0,0,0,0,1,0
%N A014082 Number of occurrences of '111' in binary expansion of n.
%C A014082 a(n) = A213629(n,7) for n > 6. - _Reinhard Zumkeller_, Jun 17 2012
%H A014082 Reinhard Zumkeller, <a href="/A014082/b014082.txt">Table of n, a(n) for n = 0..10000</a>
%H A014082 Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A014082 Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>
%H A014082 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigitBlock.html">Digit Block</a>
%H A014082 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A014082 a(2n) = a(n), a(2n+1) = a(n) + [n congruent to 3 mod 4]. - _Ralf Stephan_, Aug 21 2003
%F A014082 G.f.: 1/(1-x) * Sum_{k>=0} t^7(1-t)/(1-t^8), where t=x^2^k. - _Ralf Stephan_, Sep 08 2003
%p A014082 See A014081.
%p A014082 f:= proc(n) option remember;
%p A014082   if n::even then procname(n/2)
%p A014082   elif n mod 8 = 7 then 1 + procname((n-1)/2)
%p A014082   else procname((n-1)/2)
%p A014082 fi
%p A014082 end proc:
%p A014082 f(0):= 0:
%p A014082 map(f, [$0..1000]); # _Robert Israel_, Sep 11 2015
%t A014082 f[n_] := Count[ Partition[ IntegerDigits[n, 2], 3, 1], {1, 1, 1}]; Table[f@n, {n, 0, 104}] (* _Robert G. Wilson v_, Apr 02 2009 *)
%t A014082 a[0] = a[1] = 0; a[n_] := a[n] = If[EvenQ[n], a[n/2], a[(n - 1)/2] + Boole[Mod[(n - 1)/2, 4] == 3]]; Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Oct 22 2012, after _Ralf Stephan_ *)
%t A014082 Table[SequenceCount[IntegerDigits[n,2],{1,1,1},Overlaps->True],{n,0,110}] (* _Harvey P. Dale_, Mar 05 2023 *)
%o A014082 (Haskell)
%o A014082 import Data.List (tails, isPrefixOf)
%o A014082 a014082 = sum . map (fromEnum . ([1,1,1] `isPrefixOf`)) .
%o A014082                     tails . a030308_row
%o A014082 -- _Reinhard Zumkeller_, Jun 17 2012
%o A014082 (PARI) a(n) = hammingweight(bitand(n, bitand(n>>1, n>>2))); \\ _Gheorghe Coserea_, Aug 30 2015
%Y A014082 Cf. A014081, A033264, A056974, A056975, A056976, A056977, A056978, A056979, A056980, A213629, A239906, A239907.
%K A014082 nonn,easy,base
%O A014082 0,16
%A A014082 _Simon Plouffe_