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A047778
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Concatenation of first n numbers in binary.
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24
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1, 6, 27, 220, 1765, 14126, 113015, 1808248, 28931977, 462911642, 7406586283, 118505380540, 1896086088653, 30337377418462, 485398038695407, 15532737238253040, 497047591624097297, 15905522931971113522
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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FORMULA
| a(n) =a(n-1)*2^(1+floor[log2(n)])+n - Henry Bottomley (se16(AT)btinternet.com), Jan 12 2001
a(n) = 4C / 2^frac(log_2(n)) * n^{n+1} / r(frac(log_2(n)))^n + O(1), where r(x) = 2^{x - 1 + 2^{1-x}}; frac is the fractional part function frac(x) = x - floor(x); and C is the binary Champernowne constant (A066716). (In fact, a(n) is the floor of this expression; the error term is between 1/2 and 1.) r(x) takes on values between e*log(2) and 2 for x in the range 0 to 1. It follows using Stirling's approximation that the radius of convergence for the e.g.f. is log 2. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Sep 07 2006
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EXAMPLE
| a(4) = 1 10 11 100 = 220
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MATHEMATICA
| If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 2]]]; Table[AppendTo[n, IntegerDigits[w, 2]]; n=Flatten[n]; FromDigits[n, 2], {w, STARTPOINT, ENDPOINT}] [From Dylan Hamilton (Phalarisbull(AT)gmail.com), Aug 04 2010]
f[n_] := FromDigits[ Flatten@ IntegerDigits[ Range@n, 2], 2]; Array[f, 18] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 07 2010]
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CROSSREFS
| Cf. A001855 (bit counts, offset by 1), A061168, A066716, A007908.
Concatenation of first n numbers in other bases: 2: this sequence, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447 [From Dylan Hamilton (Phalarisbull(AT)gmail.com), Aug 11 2010]
Sequence in context: A144013 A092854 A060977 * A164985 A048436 A006174
Adjacent sequences: A047775 A047776 A047777 * A047779 A047780 A047781
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KEYWORD
| easy,nonn,base,nice
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AUTHOR
| Aaron Gulliver (gulliver(AT)elec.canterbury.ac.nz)
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EXTENSIONS
| More terms from Patrick De Geest (pdg(AT)worldofnumbers.com), May 15 1999.
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