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Let cn(n,k) denote the number of times 11..1 (k 1's) appears in the binary representation of n; a(n) = n - cn(n,1) + cn(n,2) - cn(n,3) + cn(n,4) - ... .
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%I #47 Apr 27 2024 03:33:59

%S 0,0,1,2,3,3,5,5,7,7,8,9,11,11,12,13,15,15,16,17,18,18,20,20,23,23,24,

%T 25,26,26,28,28,31,31,32,33,34,34,36,36,38,38,39,40,42,42,43,44,47,47,

%U 48,49,50,50,52,52,54,54,55,56,58,58,59,60,63,63,64,65,66,66,68,68,70,70,71,72,74,74,75

%N Let cn(n,k) denote the number of times 11..1 (k 1's) appears in the binary representation of n; a(n) = n - cn(n,1) + cn(n,2) - cn(n,3) + cn(n,4) - ... .

%H Gheorghe Coserea, <a href="/A239907/b239907.txt">Table of n, a(n) for n = 0..10000</a>

%H Jon Maiga, <a href="http://sequencedb.net/s/A239907">Computer-generated formulas for A239907</a>, Sequence Machine.

%F Conjecture: a(n) = n - A329320(n) for n >= 0 (noticed by Sequence Machine). - _Mikhail Kurkov_, Oct 13 2021

%p # From A014081:

%p cn := proc(v, k) local n, s, nn, i, j, som, kk;

%p som := 0;

%p kk := convert(cat(seq(1, j = 1 .. k)), string);

%p n := convert(v, binary);

%p s := convert(n, string);

%p nn := length(s);

%p for i to nn - k + 1 do

%p if substring(s, i .. i + k - 1) = kk then som := som + 1 fi od;

%p som; end;

%p g:=n->add((-1)^i*cn(n,i),i=1..10); # assumes n < 1023

%p [seq(n+g(n), n=0..100)];

%t cn[n_, k_] := Count[Partition[IntegerDigits[n, 2], k, 1], Table[1, {k}]]; Table[n - Sum[cn[n, i], {i, 1, IntegerLength[n, 2], 2}] + Sum[cn[n, i], {i, 2, IntegerLength[n, 2], 2}], {n, 0, 78}] (* _Michael De Vlieger_, Sep 18 2015 *)

%o (PARI)

%o binruns(n) = {

%o if (n == 0, return([1, 0]));

%o my(bag = List(), v=0);

%o while(n != 0,

%o v = valuation(n,2); listput(bag, v); n >>= v; n++;

%o v = valuation(n,2); listput(bag, v); n >>= v; n--);

%o return(Vec(bag));

%o };

%o a(n) = {

%o my(v = binruns(n));

%o n - sum(i = 1, #v, if (i%2 == 0, (v[i] + 1)\2, 0))

%o };

%o vector(79, i, a(i-1)) \\ _Gheorghe Coserea_, Sep 18 2015

%Y Cf. A000120, A012081, A014082, A239904, A239906.

%K nonn,base

%O 0,4

%A _N. J. A. Sloane_, Apr 07 2014