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