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Search: id:A125186
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| A125186 |
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Number of digits 1 in all hyperbinary representations of n. A hyperbinary representation of a nonnegative integer n is a representation of n as a sum of powers of 2, each power being used at most twice. |
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+0
1
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| 0, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 4, 7, 8, 9, 8, 10, 10, 10, 7, 11, 11, 12, 9, 12, 10, 10, 5, 11, 12, 15, 12, 17, 16, 17, 11, 18, 18, 20, 15, 20, 17, 17, 9, 18, 18, 22, 16, 23, 20, 21, 12, 21, 19, 22, 14, 20, 15, 15, 6, 16, 17, 23, 17, 27, 24, 27, 16, 29, 28, 33, 23, 33, 27, 28
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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a(n)=dB(n+1,t)/dt|_{t=1}, where B(n,t) are the Stern polynomials, defined by B(0,t)=0, B(1,t)=1, B(2n,t)=tB(n,t), B(2n+1,t)=B(n+1,t)+B(n,t) for n>=1 (see S. Klavzar et al. and A125184).
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REFERENCES
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N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math., (in press).
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FORMULA
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a(0)=0, a(1)=0, a(2)=1, a(2n+1)=a(n)+a(n+1), a(4n)=2a(2n)-a(n), a(4n+2)=a(2n+2)+a(2n) for n>=1.
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EXAMPLE
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a(8)=4 because the hyperbinary representations of 8 are 200 (=2*2^2+0*2^1+0*2^0), 120 (=1*2^2+2*2^1+0*2^0), 1000 (=1*2^3+0*2^2+0*2^1+0*2^0) and 112 (=1*2^2+1*2^1+2*1^0), having a total of 0+1+1+2=4 digits 1 (see S. Klavzar et al. Table 1).
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MAPLE
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a[0]:=0: a[1]:=0: a[2]:=1: for n from 1 to 50 do a[2*n+1]:=a[n]+a[n+1]: a[4*n]:=2*a[2*n]-a[n]: a[4*n+2]:=a[2*n+2]+a[2*n] od: seq(a[n+1], n=0..100);
B:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then t*B(n/2) else B((n+1)/2)+B((n-1)/2) fi end: seq(subs(t=1, diff(B(n+1), t)), n=0..100);
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CROSSREFS
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Cf. A125184.
Sequence in context: A153112 A005350 A055037 this_sequence A140473 A115384 A131411
Adjacent sequences: A125183 A125184 A125185 this_sequence A125187 A125188 A125189
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2006
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