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A072219
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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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5
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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
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OFFSET
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1,3
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COMMENTS
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2^k_1 is smallest power of 2 that is >= n.
The Mathematica program computes the sequence for numbers 1 to 2^m. - T. D. Noe, Jul 15 2002
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REFERENCES
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P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, pp. 61-62.
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LINKS
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Table of n, a(n) for n=1..105.
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FORMULA
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G.f. 1/(1+x) + sum(r>=0, x^(2^r) / ( 1+x^(2^(r+1)) ) ) / (1-x) .- Ramasamy Chandramouli, Dec 22 2012.
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EXAMPLE
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1=1, 2=2, 3=4-2+1, 4=4, 5=8-4+1, 6=8-4+2, ...
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MATHEMATICA
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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
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CROSSREFS
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Cf. A072339, A065621.
Equals 2*A033264(n-1) + 1.
Sequence in context: A081325 A132680 A105595 * A173854 A059789 A023136
Adjacent sequences: A072216 A072217 A072218 * A072220 A072221 A072222
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Jul 05 2002
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EXTENSIONS
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More terms from T. D. Noe, Jul 15 2002
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STATUS
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approved
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