A037888 a(n) = (1/2)*Sum_{i} |d(i) - e(i)| where Sum_{i} d(i)*2^i is the base-2 representation of n and e(i) are digits d(i) in reverse order.

0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 0, 2, 1, 2, 1, 1, 0, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 2, 1, 3, 2, 1, 0, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 0, 2, 1, 2, 1, 3

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base
• nice

Comments

a(n) = least number of digits for which the change 0->1 in (binary n) yields a palindrome.

a(n) = Sum_{k=0..A070939(n)/2-1} abs(A030308(n, k) - A030308(n, A070939(n)-k)). - Reinhard Zumkeller, Apr 09 2013

a(n) = Sum_{k=0..A070939(n)/2-1} ((A030308(n, k) + A030308(n, A070939(n)-k)) mod 2). - Reinhard Zumkeller, Sep 18 2013

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A064834.

Programs

Haskell

a037888 n = div (sum $ map abs $ zipWith (-) bs $ reverse bs) 2
   where bs = a030308_row n
-- Reinhard Zumkeller, Apr 09 2013

Maple

a:= proc(n) local r, ad: r:= proc(s) options operator, arrow: [seq(s[nops(s)-j+1], j = 1 .. nops(s))] end proc: ad := proc(s) local i,j: j := 0: for i to nops(s) do if 0 < abs((s-r(s))[i]) then j := j+1 else end if end do: (1/2)*j end proc: ad(convert(n, base, 2)) end proc: seq(a(n), n = 1 .. 90); # Emeric Deutsch, Aug 20 2016

Mathematica

a[n_] := (bits = IntegerDigits[n, 2]; Total[Abs[bits - Reverse[bits]]]/2); Table[a[n], {n, 1, 90}] (* Jean-François Alcover, Jan 16 2013 *)

PARI

for(n = 1, 90,
  v = binary(n); s = 0; j = #v;
  for(k=1,#v, s+=abs(v[k]-v[j]); j--);
  s/=2;
  print1(s,", ")
)
\\ Washington Bomfim, Jan 13 2011