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A029931
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If 2n = Sum 2^e_i, a(n) = Sum e_i.
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15
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0, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 5, 6, 7, 8, 8, 9, 10, 11, 9, 10, 11, 12, 12, 13, 14, 15, 6, 7, 8, 9, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 16, 11, 12, 13, 14, 14, 15, 16, 17, 15, 16, 17, 18, 18, 19, 20, 21, 7, 8, 9, 10, 10, 11, 12, 13, 11, 12, 13, 14, 14, 15, 16
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Write n in base 2, n = sum b(i)*2^(i-1), then a(n) = sum b(i)*i - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 09 2002
May be regarded as a triangular array read by rows, giving weighted sum of compositions in standard order. The standard order of compositions is given by A066099. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 06 2006
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1023
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197, ex. 10.
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FORMULA
| a(n) =a(n-2^L(n))+L(n)+1 [where L(n)=floor[log_2(n)]=A000523(n)] = sum of digits of A048794 [at least for n<512] - Henry Bottomley (se16(AT)btinternet.com), Mar 09 2001
a(1)=0, a(2n) = a(n)+e1(n), a(2n+1) = a(2n)+1, where e1(n) = A000120(n). a(n) = log2(A029930(n)). - Ralf Stephan, Jun 19 2003
G.f. 1/(1-x) * sum(k>=0, (k+1)*x^2^k/(1+x^2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 23 2003
a(n)=Sum_k>=0 {A030308(n,k)*A000027(k+1)}. - From DELEHAM Philippe, Oct 15 2011.
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EXAMPLE
| 14 = 8+4+2 so a(7) = 3+2+1 = 6.
Composition number 11 is 2,1,1; 1*2+2*1+3*1 = 7, so a(11) = 7.
The triangle starts:
0
1
2 3
3 4 5 6
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MATHEMATICA
| a[n_] := (b = IntegerDigits[n, 2]).Reverse @ Range[Length @ b]; Array[a, 78, 0] (* From Jean-François Alcover, Apr 28 2011, after B. Cloitre *)
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PROG
| (PARI) for(n=0, 100, l=length(binary(n)); print1(sum(i=1, l, component(binary(n), i)*(l-i+1)), ", "))
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CROSSREFS
| Cf. A059867, A073642.
Other sequences that are built by replacing 2^k in the binary representation with other numbers: A089625 (primes), A059590 (factorials), A022290 (Fibonacci).
Cf. A066099, A070939, A124757, A011782 (row lengths), A001793 (row sums).
Sequence in context: A156562 A007998 A202704 * A022290 A185363 A103827
Adjacent sequences: A029928 A029929 A029930 * A029932 A029933 A029934
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KEYWORD
| nonn,easy,nice,tabf
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Erich Friedman (erich.friedman(AT)stetson.edu).
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