|
| |
|
|
A181132
|
|
a(0)=0; thereafter a(n) = total number of 0's in binary expansions of 1, ..., n.
|
|
4
|
|
|
|
0, 0, 1, 1, 3, 4, 5, 5, 8, 10, 12, 13, 15, 16, 17, 17, 21, 24, 27, 29, 32, 34, 36, 37, 40, 42, 44, 45, 47, 48, 49, 49, 54, 58, 62, 65, 69, 72, 75, 77, 81, 84, 87, 89, 92, 94, 96, 97, 101, 104, 107, 109, 112, 114, 116, 117, 120, 122, 124, 125, 127, 128, 129, 129, 135, 140, 145
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
The graph of this sequence is a version of the Takagi curve: see Lagarias (2012), Section 9, especially Theorem 9.1. - N. J. A. Sloane, Mar 12 2016
|
|
|
LINKS
|
Jeffrey C. Lagarias, The Takagi function and its properties, In Functions in number theory and their probabilistic aspects, 153--189, RIMS Kôkyûroku Bessatsu, B34, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012. MR3014845.
|
|
|
FORMULA
|
a(0)=0; thereafter a(2n) = a(n)+a(n-1)+n, a(2n+1) = 2a(n)+n. G.f.: (1/(1-x)^2) * Sum_{k >= 0} x^(2^(k+1))/(1+x^(2^k)). - N. J. A. Sloane, Mar 10 2016
|
|
|
MATHEMATICA
|
Accumulate[Count[IntegerDigits[#, 2], 0]&/@Range[70]] (* Harvey P. Dale, May 16 2012 *)
Join[{0}, Accumulate[DigitCount[Range[70], 2, 0]]] (* Harvey P. Dale, Jun 09 2016 *)
|
|
|
PROG
|
(Other) microsoft word macro: Sub concatcount() Dim base As Integer Dim digit As Integer Dim counter As Integer Dim standin As Integer Dim max As Integer Let base = 2 Let digit = 0 Let max = 100 Let counter = 0 For n = 1 To max Let standin = n Do While standin > 0 If standin Mod base = digit Then Let counter = counter + 1 Let standin = standin - (standin Mod base) If standin > 0 Then Let standin = standin / base Loop Selection.TypeText Text:=Str(counter) Next n End Sub
(Magma) [ n eq 1 select 0 else Self(n-1)+(#B-&+B) where B is Intseq(n, 2): n in [1..70] ]; // Klaus Brockhaus, Oct 08 2010
(PARI) a(n)=my(m=logint(n, 2)); 1 + (m+1)*(n+1) - 2^(m+1) + sum(j=1, m+1, my(t=floor(n/2^j + 1/2)); (n>>j)*(2*n + 2 - (1 + (n>>j))<<j) - (2*n + 2 - t<<j)*t)/2 \\ Charles R Greathouse IV, Dec 14 2015
(Python)
def A181132(n): return 1+(n+1)*(m:=(n+1).bit_length())-(1<<m)-sum(i.bit_count() for i in range(1, n+1)) # Chai Wah Wu, Mar 02 2023
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
base,easy,nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
a(0)=0 added by N. J. A. Sloane, Mar 10 2016 (simplifies the recurrence, makes entry consistent with A059015 and other closely related sequences).
|
|
|
STATUS
|
approved
|
| |
|
|
