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A268289 a(0)=0; thereafter a(n) = a(n-1) - A037861(n). 4
0, 1, 1, 3, 2, 3, 4, 7, 5, 5, 5, 7, 7, 9, 11, 15, 12, 11, 10, 11, 10, 11, 12, 15, 14, 15, 16, 19, 20, 23, 26, 31, 27, 25, 23, 23, 21, 21, 21, 23, 21, 21, 21, 23, 23, 25, 27, 31, 29, 29, 29, 31, 31, 33, 35, 39, 39, 41, 43, 47, 49 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The graph of this sequence is a version of the Takagi curve: see Lagarias (2012), Section 9, especially Theorem 9.1.

REFERENCES

Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

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. Also arxiv preprint, arXiv:1112.4502 [math.CA], 2011-2012.

FORMULA

From N. J. A. Sloane, Mar 11 2016: (Start)

a(0)=0; for n > 0, a(n) = a(n-1) + A000120(n) - A023416(n) = A000788(n) - A181132(n).

a(0)=0; thereafter a(2n) = a(n) + a(n-1), a(2n+1) = 2a(n) + 1.

G.f.: (1/(1-x)^2) * Sum_{k >= 0} x^(2^k)*(1-x^(2^k))/(1+x^(2^k)).

a(2^k-1) = 2^k-1, a(3*2^k-1) = 2^(k+1)-1, a(5*2^k-1) = 3*2^k-1, etc.

(End)

MAPLE

# Maple code from N. J. A. Sloane, Mar 07 2016

a000120 := proc(n) add(i, i=convert(n, base, 2)) end:

a023416 := proc(n) if n = 0 then 1; else add(1-e, e=convert(n, base, 2)) ; end if; end proc:

a268289:=proc(n) option remember; global a000120, a023416;

if n=0 then 0 else a268289(n-1)+a000120(n)-a023416(n); fi; end;

[seq(a268289(n), n=0..132)];

MATHEMATICA

Join[{0}, Table[DigitCount[n, 2, 1] - DigitCount[n, 2, 0], {n, 1, 100}] // Accumulate] (* Jean-François Alcover, Oct 24 2016 *)

PROG

(Python)

def sequence(x):

x = str(bin(x))

y = []

for z in list(x):

  if z == '0':

   y.append(-1)

  else:

   y.append(1)

y.pop(0)

y.pop(0)

return(y)

#Use for sequence

def pattern(upto):

p = []

seed = 0

adds = []

while upto != 0:

  adds.append(sequence(upto))

  upto -= 1

while len(adds) != 0:

  x = adds.pop()

  for y in x:

   seed += y

  p.append(seed)

return(p)

#Use for numbers in / not in sequence

def notin(upto, inn):

x = pattern(int(upto) ** 2)

y = []

while upto != 0:

  if inn == False:

   if upto not in x:

    y.append(upto)

  else:

   if upto in x:

    y.append(upto)

  upto -= 1

y.reverse()

return(y)

(PARI) a(n) = if (n==0, 0, a(n-1) + 2* hammingweight(n) - #binary(n)); \\ Michel Marcus, Jan 31 2016

(PARI) lista(nn) = {a = 0; for (n=1, nn, a += 2* hammingweight(n) - #binary(n); print1(a, ", "); ); }

CROSSREFS

Cf. A000788, A181132, A037861, A023416, A000120, A269735.

Sequence in context: A132408 A091821 A086035 * A201907 A003559 A200599

Adjacent sequences:  A268286 A268287 A268288 * A268290 A268291 A268292

KEYWORD

nonn,base,hear,nice

AUTHOR

Mark Moore, Jan 30 2016

EXTENSIONS

Simplified definition following a suggestion from Michel Marcus. Corrected start, added more terms. - N. J. A. Sloane, Mar 07 2016

STATUS

approved

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Last modified February 24 22:41 EST 2018. Contains 299627 sequences. (Running on oeis4.)