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A256696 R(k), the minimal alternating binary representation of k, concatenated for k = 0, 1, 2,.... 7
0, 1, 2, 4, -1, 4, 8, -4, 1, 8, -2, 8, -1, 8, 16, -8, 1, 16, -8, 2, 16, -8, 4, -1, 16, -4, 16, -4, 1, 16, -2, 16, -1, 16, 32, -16, 1, 32, -16, 2, 32, -16, 4, -1, 32, -16, 4, 32, -16, 8, -4, 1, 32, -16, 8, -2, 32, -16, 8, -1, 32, -8, 32, -8, 1, 32, -8, 2, 32 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Suppose that b = (b(0), b(1), ... ) is an increasing sequence of positive integers satisfying b(0) = 1 and b(n+1) <= 2*b(n) for n >= 0.  Let B(n) be the least b(m) >= n.  Let R(0) = 1, and for n > 0, let R(n) = B(n) - R(B(n) - n).  The resulting sum of the form R(n) = B(n) - B(m(1)) + B(m(2)) - ... + ((-1)^k)*B(k) is the minimal alternating b-representation of n.  The sum B(n) + B(m(2)) + ... is the positive part of R(n), and the sum B(m(1)) + B(m(3)) + ... , the nonpositive part of R(n).  The number ((-1)^k)*B(k) is the trace of n.

If b(n) = 2^n, the sum R(n) is the minimal alternating binary representation of n.

A055975 = trace of n, for n >= 1.

A091072 gives the numbers having positive trace.

A091067 gives the numbers having negative trace.

A072339 = number of terms in R(n).

A073122 = sum of absolute values of the terms in R(n).

REFERENCES

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1981, Vol. 2 (2nd ed.), p. 196, Exercise 27.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

EXAMPLE

R(0) = 0

R(1) = 1

R(2) = 2

R(3) = 4 - 1

R(4) = 4

R(9) = 8 - 4 + 1

R(11) = 16 - 8 + 4 - 1

MATHEMATICA

z = 100; b[n_] := 2^n; bb = Table[b[n], {n, 0, 40}];

s[n_] := Table[b[n + 1], {k, 1, b[n]}];

h[0] = {1}; h[n_] := Join[h[n - 1], s[n - 1]];

g = h[10]; r[0] = {0};

r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]]

u = Flatten[Table[r[n], {n, 0, z}]]

CROSSREFS

Cf. A000072, A256655 (Fibonacci based), A055975, A091072, A091067, A072339, A073122, A256701, A256702.

Sequence in context: A135185 A201774 A011029 * A244261 A085111 A181332

Adjacent sequences:  A256693 A256694 A256695 * A256697 A256698 A256699

KEYWORD

easy,sign,base

AUTHOR

Clark Kimberling, Apr 09 2015

STATUS

approved

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Last modified April 26 11:44 EDT 2017. Contains 285444 sequences.