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A065359 Alternating bit sum for n: replace 2^k with (-1)^k in binary expansion of n. 20
0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2, -1, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2, -1, 0, 1, -1, 0, -2, -1, -3, -2, -1, 0, -2, -1, 0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2, -1, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Notation: (2)[n](-1)

Comments from David W. Wilson and Ralf Stephan, Jan 09 2007: (Start)

a(n) is even iff n in A001969; a(n) is odd iff n in A000069.

a(n) == 0 (mod 3) iff n == 0 (mod 3).

a(n) == 0 (mod 6) iff (n == 0 (mod 3) and n/3 not in A036556).

a(n) == 3 (mod 6) iff (n == 0 (mod 3) and n/3 in A036556). (End)

a(n) = A030300(n) - A083905(n). - Ralf Stephan, Jul 12 2003

First occurrence of k and -k: 0, 1, 2, 5, 10, 21, 42, 85, ..., (A000975), i.e.; first 0 occurs for 0, first 1 occurs for 1, first -1 occurs at 2, first 2 occurs for 5, etc....,

a(n)=-3 only if mod(n,3)=0,

a(n)=-2 only if mod(n,3)=1,

a(n)=-1 only if mod(n,3)=2,

a(n)= 0 only if mod(n,3)=0,

a(n)= 1 only if mod(n,3)=1,

a(n)= 2 only if mod(n,3)=2,

a(n)= 3 only if mod(n,3)=0, ..., .

a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - From Philippe Deléham, Oct 20 2011.

REFERENCES

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.

LINKS

Harry J. Smith, Table of n, a(n) for n=0,...,1000

J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II

William Paulsen, wpaulsen(AT)csm.astate.edu, Partitioning the [prime] maze

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

FORMULA

G.f.: 1/(1-x) * sum(k>=0, (-1)^k*x^2^k/(1+x^2^k)). - Ralf Stephan, Mar 07 2003

a(0) = 0, a(2n) = -a(n), a(2n+1) = 1-a(n). - Ralf Stephan, Mar 07 2003

a(n)=Sum_k>=0 {A030308(n,k)*(-1)^k}. - From Philippe Deléham, Oct 20 2011.

EXAMPLE

Alternating bit sum for 11 = 1011 in binary is 1 - 1 + 0 - 1 = -1, so a(11) = -1.

MAPLE

A065359 := proc(n) local dgs ; dgs := convert(n, base, 2) ; add( -op(i, dgs)*(-1)^i, i=1..nops(dgs)) ; end proc: # R. J. Mathar, Feb 04 2011

MATHEMATICA

f[0]=0; f[n_] := Plus @@ (-(-1)^Range[ Floor[ Log2@ n + 1]] Reverse@ IntegerDigits[n, 2]); Array[ f, 107, 0]

PROG

(PARI) SumAD(x)= { local(a=1, s=0); while (x>9, s+=a*(x-10*(x\10)); x\=10; a=-a); return(s + a*x) } baseE(x, b)= { local(d, e=0, f=1); while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) } { for (n=0, 1000, b=baseE(n, 2); write("b065359.txt", n, " ", SumAD(b)) ) } [From Harry J. Smith, Oct 17 2009]

(PARI) for(n=0, 106, s=0; u=1; for(k=0, #binary(n)-1, s+=bittest(n, k)*u; u=-u); print1(s, ", ")) /* W. Bomfim, Jan 18 2011 */

CROSSREFS

Cf. A000120, A065360, A065368, A065081.

Partial sums seem to be in A005536.

Sequence in context: A117355 A086966 A140080 * A087372 A036431 A029407

Adjacent sequences:  A065356 A065357 A065358 * A065360 A065361 A065362

KEYWORD

base,easy,sign

AUTHOR

Marc LeBrun, Oct 31 2001

EXTENSIONS

More terms from Ralf Stephan, Jul 12 2003

STATUS

approved

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Last modified April 16 21:07 EDT 2014. Contains 240627 sequences.