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A306607 The bottom entry in the difference table of the binary digits of n. 3
0, 1, 1, 0, 1, 2, -1, 0, 1, 0, 4, 3, -2, -3, 1, 0, 1, 2, -3, -2, 7, 8, 3, 4, -3, -2, -7, -6, 3, 4, -1, 0, 1, 0, 6, 5, -9, -10, -4, -5, 11, 10, 16, 15, 1, 0, 6, 5, -4, -5, 1, 0, -14, -15, -9, -10, 6, 5, 11, 10, -4, -5, 1, 0, 1, 2, -5, -4, 16, 17, 10, 11, -19 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

By convention, a(0) = 0.

For any n > 0: let (b_0, ..., b_w) be the binary representation of n:

- b_w = 1, and for any k = 0..w, 0 <= b_k <= 1,

- n = Sum_{k = 0..w} b_k * 2^k,

- a(n) is the unique value remaining after taking successively the first differences of (b_0, ..., b_w) w times.

From Robert Israel, Mar 07 2019: (Start)

  If n is odd then f(A030101(n)) = (-1)^A000523(n)*f(n).

  In particular, if n is in A048701 then a(n)=0.

  a(n) == 1 (mod A014963(A000523(n))) if n is even,

  a(n) == 0 (mod A014963(A000523(n))) if n is odd. (End)

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

FORMULA

a(2^k) = 1 for any k >= 0.

a(2^k-1) = 0 for any k > 1.

a(3*2^k) = -k for any k >= 0.

a(n) = Sum_{k=0..A000523(n)} binomial(A000523(n), k)*(-1)^k*A030302(n,k). - David A. Corneth, Mar 07 2019

G.f.: 1/(x-1)*Sum_{k>=0}(x^(2^(k+1))-x^(2^k) + x^(2^k)/(x^(2^k)+1)*Sum_{m>=k+1}(binomial(m,k)*(-1)^(m-k)*(x^(2^(m+1))-x^(2^m)))). - Robert Israel, Mar 07 2019

EXAMPLE

For n = 42:

- the binary representation of 42 is "101010",

- the corresponding difference table is:

   0   1   0   1   0   1

     1  -1   1  -1   1

      -2   2  -2   2

         4  -4   4

          -8   8

            16

- hence a(42) = 16.

MAPLE

f:= proc(n) local L;

  L:= convert(n, base, 2);

  while nops(L) > 1 do

    L:= L[2..-1]-L[1..-2]

  od;

  op(L)

end proc:

map(f, [$0..100]); # Robert Israel, Mar 07 2019

MATHEMATICA

a[n_] := NestWhile[Differences, Reverse[IntegerDigits[n, 2]], Length[#] > 1 &][[1]]; Array[a, 100, 0] (* Amiram Eldar, Mar 08 2019 *)

PROG

(PARI) a(n) = if (n, my (v=Vecrev(binary(n))); while (#v>1, v=vector(#v-1, k, (v[k+1]-v[k]))); v[1], 0)

(PARI) a(n) = my(b = binary(n), s = -1); sum(i = 1, #b, s=-s; binomial(#b-1, i-1) * b[i] * s) \\ David A. Corneth, Mar 07 2019

CROSSREFS

Cf. A000523, A007088, A030101, A030190, A030302, A048701, A014963, A187202, A241494.

Sequence in context: A072943 A072175 A280712 * A268611 A092147 A208134

Adjacent sequences:  A306604 A306605 A306606 * A306608 A306609 A306610

KEYWORD

sign,base,look

AUTHOR

Rémy Sigrist, Feb 28 2019

STATUS

approved

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Last modified August 22 12:34 EDT 2019. Contains 326177 sequences. (Running on oeis4.)