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A117127
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Concatenate the first n positive integers written in binary (with each integer written so the most significant 1 is on the left and the concatenated string is from left to right). a(n) is the number of times n written in binary appears in the concatenated string.
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3
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1, 1, 2, 1, 2, 3, 3, 1, 2, 1, 4, 3, 4, 5, 4, 1, 2, 2, 3, 2, 3, 2, 7, 3, 4, 4, 6, 6, 5, 7, 5, 1, 2, 2, 3, 1, 4, 3, 4, 2, 4, 2, 5, 2, 4, 4, 9, 3, 4, 5, 5, 4, 6, 4, 9, 6, 6, 6, 7, 9, 6, 9, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 2, 6, 4, 5, 2, 4, 3, 6, 3, 5, 4, 7, 2, 5, 4, 7, 3, 7, 5, 11, 3, 4, 5, 5, 4, 7, 5, 7, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| See also A047778, concatenation of first n numbers in binary. - Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 21 2006
Note that these counts reflect the pattern overlapping case. - Diana Mecum (diana.mecum(AT)gmail.com), Aug 06 2008
First occurrence of n: 1, 3, 6, 11, 14, 27, 23, 107, 47, 119, 95, 111, 191, 239, 223, 1007, 767, 447, 1535, 991, 895, 959, 3519, 1791, 1983, 1919, 3583, 7159, 7871, 3839, 14327, 16063, 8063, 7679, 7935, 15839, ??, 15359, 16127, 15871, ..., ; limit of search = 25000. - Robert G. Wilson v, Aug 30 2008.
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LINKS
| Diana Mecum and RGWv, Table of n, a(n) for n = 1..10000
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EXAMPLE
| The first 13 positive integers written in binary and concatenated is
11011100101110111100010011010101111001101.
13 in binary, which is 1101, appears 4 times in the concatenated string, starting at positions (reading from left to right) 1, 12, 25 and 38.
The last occurrence of 1101 is the integer 13 itself converted to binary and added to the end of the concatenated string, of course. So a(13) = 4.
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MAPLE
| A007088 := proc(n) convert(n, base, 2) ; ListTools[Reverse](%) ; end:
A058935 := proc(n) local i, a ; if n = 0 then RETURN([0]) ; else a := [] ; for i from 1 to n do a := [op(a), op(A007088(i))] ; od: fi ; end:
A117127 := proc(n) local a058935, strtL, endL, nL, a ; nL := A007088(n) ; a058935 := A058935(n) ; a := 0 ; for strtL from 1 to nops(a058935) do for endL from strtL to nops(a058935) do if verify[sublist]( nL, [op(strtL..endL, a058935)] ) then a := a+1 ; fi ; od: od ; RETURN(a) ; end: for n from 1 to 80 do printf("%d, ", A117127(n)) ; od: - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 23 2008
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MATHEMATICA
| a[n_] := StringCount[ ToString@ FromDigits@ Flatten@ IntegerDigits[ Range@n, 2], ToString@ FromDigits@ IntegerDigits[n, 2], Overlaps -> True]; Array[a, 105] (* Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 30 2008 *)
fc[n_]:=Module[{idn2=IntegerDigits[n, 2], len}, len=Length[idn2]; Count[ Partition[ Flatten[Table[IntegerDigits[i, 2], {i, n}]], len, 1], idn2]]; Array[fc, 110] (* From Harvey P. Dale, Dec 16 2011 *)
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CROSSREFS
| Cf. A007088, A030190, A058935, A047778, A141439.
Sequence in context: A140575 A186333 A101933 * A136624 A033763 A033803
Adjacent sequences: A117124 A117125 A117126 * A117128 A117129 A117130
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KEYWORD
| easy,nonn,base
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AUTHOR
| Leroy Quet Apr 20 2006
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 23 2008
Terms 81 through 2000 from Diana Mecum (diana.mecum(AT)gmail.com), Aug 06 2008
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