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.

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

Author

N. J. A. Sloane, Jul 05 2002

Keywords

• nonn
• easy
• nice

Comments

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

Examples

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

References

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

Links

• 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

Crossrefs

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

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

Programs

Haskell

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

Mathematica

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 *)

Formula

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

Extensions

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