A072339 -id:A072339 - cp's OEIS frontend

A073122 Minimal reversing binary representation of n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives n. See A072339.

1, 2, 5, 4, 13, 10, 9, 8, 25, 26, 29, 20, 21, 18, 17, 16, 49, 50, 53, 52, 61, 58, 57, 40, 41, 42, 45, 36, 37, 34, 33, 32, 97, 98, 101, 100, 109, 106, 105, 104, 121, 122, 125, 116, 117, 114, 113, 80, 81, 82, 85, 84, 93, 90, 89, 72, 73, 74, 77, 68, 69, 66, 65, 64, 193

Offset: 1

T. D. Noe, Jul 17 2002

The minimal representation is unique. The number of powers of 2 can be either even or odd. Compare with A065621, in which the number of powers of 2 is odd. The Mathematica program computes the representation for numbers 1 to 2^m. a(0) = 0.

No term has odd part congruent to 3 modulo 4. - Charlie Neder, Oct 28 2018

			a(11) = 29 because 29 = 16 + 8 + 4 + 1 and 16 - 8 + 4 - 1 = 11.
a(100) = 164 because 100 in binary is 1100100. The two runs of ones correspond to 2^7 - 2^5 and 2^3 - 2^2, but since 2^3 - 2^2 is the last term of the representation, it can be replaced with 2^2. Therefore, a(100) = 2^7 + 2^5 + 2^2. - _Charlie Neder_, Oct 28 2018

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1981, Vol. 2 (Second Edition), p. 196, (exercise 4.1. Nr. 27)

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A065621, A072219, A072339, A256696.

Needs["DiscreteMath`Combinatorica`"]; sumit[s_List] := Module[{i, ss=0}, Do[If[OddQ[i], ss+=s[[ -i]], ss-=s[[ -i]]], {i, Length[s]}]; ss]; m=7; powers=Table[2^i, {i, 0, m}]; lst=Table[2m, {2^m}]; lst2=Table[0, {2^m}]; Do[t=NthSubset[i, powers]; len=Length[t]; st=sumit[t]; If[len

a(2n) = 2 * a(n). [Corrected by Sean A. Irvine, Nov 17 2024]

Express n as a sum of terms 2^x - 2^y, x > y, such that each term defines a run of 1's in n's binary expansion. Then a(n) is the sum of all 2^x + 2^y, with the exception that a term 2^(x+1) - 2^x at the end of a representation becomes 2^x. - Charlie Neder, Oct 28 2018

A072219 Any number n can be written uniquely in the form n = 2^k_1 - 2^k_2 + 2^k_3 - ... + 2^k_{2r+1} where the signs alternate, there are an odd number of terms, and k_1 > k_2 > k_3 > ... > k_{2r+1} >= 0; sequence gives number of terms 2r+1.

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 3, 1, 3, 3, 5, 3, 3, 3, 3, 1, 3, 3, 5, 3, 5, 5, 5, 3, 3, 3, 5, 3, 3, 3, 3, 1, 3, 3, 5, 3, 5, 5, 5, 3, 5, 5, 7, 5, 5, 5, 5, 3, 3, 3, 5, 3, 5, 5, 5, 3, 3, 3, 5, 3, 3, 3, 3, 1, 3, 3, 5, 3, 5, 5, 5, 3, 5, 5, 7, 5, 5, 5, 5, 3, 5, 5, 7, 5, 7, 7, 7, 5, 5, 5, 7, 5, 5, 5, 5, 3, 3, 3, 5, 3, 5, 5, 5, 3, 5

Offset: 1

N. J. A. Sloane, Jul 05 2002

2^k_1 is smallest power of 2 that is >= n.
The first Mathematica program computes the sequence for numbers 1 to 2^m. - T. D. Noe, Jul 15 2002
a(A000079(n)) = 1; a(A238246(n)) = 3; a(A238247(n)) = 5; a(A238248(n)) = 7. - Reinhard Zumkeller, Feb 20 2014
Add 1 to every other terms of A005811. - N. J. A. Sloane, Jan 14 2017

			1=1, 2=2, 3=4-2+1, 4=4, 5=8-4+1, 6=8-4+2, ...

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, pp. 61-62.

Reinhard Zumkeller, Table of n, a(n) for n = 1..16384
S. Kropf and S. Wagner, q-Quasiadditive functions, arXiv:1605.03654 [math.CO], 2016. See section "The number of runs and the Gray code".
Index entries for sequences related to binary expansion of n

Cf. A000079, A005811, A030308, A072339, A065621, A238246, A238247, A238248.

Equals 2*A033264(n-1) + 1.

a072219 = (+ 1) . (* 2) . a033264 . subtract 1
-- Reinhard Zumkeller, Feb 20 2014

Needs["DiscreteMath`Combinatorica`"]; sumit[s_List] := Module[{i, ss=0}, Do[If[OddQ[i], ss+=s[[i]], ss-=s[[i]]], {i, Length[s]}]; ss]; m=8; powers=Table[2^i, {i, 0, m}]; lst=Table[0, {2^m}]; sets={}; Do[sets=Union[sets, KSubsets[powers, i]], {i, 1, m+1, 2}]; Do[t=sets[[i]]; lst[[sumit[t]]]=Length[t], {i, Length[sets]}]; lst
(* second program *)
a[n_] := 2 Count[Split[IntegerDigits[n-1, 2], #1 == 1 && #2 == 0 &], {1, 0} ] + 1; Array[a, 105] (* Jean-François Alcover, Apr 01 2016 *)

G.f.: 1/(1+x) + (1/(1-x)) * Sum_{r>=0} x^(2^r) / (1+x^(2^(r+1))). - Ramasamy Chandramouli, Dec 22 2012

More terms from T. D. Noe, Jul 15 2002

A256696 R(k), the minimal alternating binary representation of k, concatenated for k = 0, 1, 2,....

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Apr 09 2015

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).

			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

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

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

Cf. A000079, A256655 (Fibonacci based), A055975, A091072, A091067, A072339, A073122, A256701, A256702.

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}]]

cp's OEIS Frontend

A073122 Minimal reversing binary representation of n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives n. See A072339.

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A072219 Any number n can be written uniquely in the form n = 2^k_1 - 2^k_2 + 2^k_3 - ... + 2^k_{2r+1} where the signs alternate, there are an odd number of terms, and k_1 > k_2 > k_3 > ... > k_{2r+1} >= 0; sequence gives number of terms 2r+1.

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A256696 R(k), the minimal alternating binary representation of k, concatenated for k = 0, 1, 2,....

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