A239907 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) - ... .

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

Offset: 0

Author

N. J. A. Sloane, Apr 07 2014

Keywords

• nonn
• base

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..10000
• Jon Maiga, Computer-generated formulas for A239907, Sequence Machine.

Crossrefs

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

Programs

Maple

# From A014081:
cn := proc(v, k) local n, s, nn, i, j, som, kk;
som := 0;
kk := convert(cat(seq(1, j = 1 .. k)), string);
n := convert(v, binary);
s := convert(n, string);
nn := length(s);
for i to nn - k + 1 do
if substring(s, i .. i + k - 1) = kk then som := som + 1 fi od;
som; end;
g:=n->add((-1)^i*cn(n,i),i=1..10); # assumes n < 1023
[seq(n+g(n), n=0..100)];

Mathematica

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 *)

PARI

binruns(n) = {
  if (n == 0, return([1, 0]));
  my(bag = List(), v=0);
  while(n != 0,
        v = valuation(n,2); listput(bag, v); n >>= v; n++;
        v = valuation(n,2); listput(bag, v); n >>= v; n--);
  return(Vec(bag));
};
a(n) = {
  my(v = binruns(n));
  n - sum(i = 1, #v, if (i%2 == 0, (v[i] + 1)\2, 0))
};
vector(79, i, a(i-1))  \\ Gheorghe Coserea, Sep 18 2015

Formula

Conjecture: a(n) = n - A329320(n) for n >= 0 (noticed by Sequence Machine). - Mikhail Kurkov, Oct 13 2021