|
%I M0469 #104 Dec 01 2022 17:47:16
%S 0,1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,2,0,2,1,
%T 2,2,2,3,2,4,2,5,2,6,2,7,2,8,2,9,3,0,3,1,3,2,3,3,3,4,3,5,3,6,3,7,3,8,
%U 3,9,4,0,4,1,4,2,4,3,4,4,4,5,4,6,4,7,4,8,4,9,5,0,5,1,5,2,5,3,5,4,5,5,5,6,5,7
%N The almost-natural numbers: write n in base 10 and juxtapose digits.
%C Also called the Barbier infinite word.
%C This is an example of a non-morphic sequence.
%C a(n) = A162711(n,1); A136414(n) = 10*a(n) + a(n+1). - _Reinhard Zumkeller_, Jul 11 2009
%C a(A031287(n)) = 0, a(A031288(n)) = 1, a(A031289(n)) = 2, a(A031290(n)) = 3, a(A031291(n)) = 4, a(A031292(n)) = 5, a(A031293(n)) = 6, a(A031294(n)) = 7, a(A031295(n)) = 8, a(A031296(n)) = 9. - _Reinhard Zumkeller_, Jul 28 2011
%C May be regarded as an irregular table in which the n-th row lists the digits of n. - _Jason Kimberley_, Dec 07 2012
%C The digits of the integer n start at index A117804(n). The digit a(n) at index n belongs to the number A100470(n). - _M. F. Hasler_, Oct 23 2019
%C See also the Copeland-Erdős constant A033308, equivalent using primes instead of all numbers. - _M. F. Hasler_, Oct 24 2019
%C Decimal expansion of Sum_{k>=1} k/10^(A058183(k) + 1). - _Stefano Spezia_, Nov 30 2022
%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 114, 336.
%D R. Honsberger, Mathematical Chestnuts from Around the World, MAA, 2001; see p. 163.
%D M. Kraitchik, Mathematical Recreations. Dover, NY, 2nd ed., 1953, p. 49.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert G. Wilson v, <a href="/A007376/b007376.txt">Table of n, a(n) for n = 0..100000</a> (a(0) = 0 added by _M. F. Hasler_, Oct 23 2019).
%H Putnam Competition No. 48, <a href="http://www.jstor.org/stable/2690046">Problem A2</a>, Math. Mag., 61 (1988), 131-134.
%H R. G. Wilson v, <a href="/A007376/a007376.pdf">Letter to N. J. A. Sloane, Oct. 1993</a>
%p c:=proc(x,y) local s: s:=proc(m) nops(convert(m,base,10)) end: if y=0 then 10*x else x*10^s(y)+y: fi end: b:=proc(n) local nn: nn:=convert(n,base,10):[seq(nn[nops(nn)+1-i],i=1..nops(nn))] end: A:=0: for n from 1 to 75 do A:=c(A,n) od: b(A); # c concatenates 2 numbers while b converts a number to the sequence of its digits - _Emeric Deutsch_, Jul 27 2006
%p #alternative
%p A007376 := proc(n) option remember ; local aprev, dOld,N ; if n <=9 then RETURN([n,n,1]) ; else aprev := A007376(n-1) ; dOld := op(3,aprev) ; N := op(2,aprev) ; if dOld < A055642(N) then RETURN([op(-dOld-1,convert(N,base,10)),N,dOld+1]) ; else RETURN([op(-1,convert(N+1,base,10)),N+1,1]) ; fi ; fi ; end: # _R. J. Mathar_, Jan 21 2008
%t Flatten[ IntegerDigits /@ Range@ 57] (* Or *)
%t almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ almostNatural[#, 10] &, 105] (* updated Jun 29 2014 *)
%t With[{nn=120},RealDigits[N[ChampernowneNumber[],nn],10,nn]][[1]] (* _Harvey P. Dale_, Mar 13 2018 *)
%o (Haskell)
%o a007376 n = a007376_list !! (n-1)
%o a007376_list = concatMap (map (read . return) . show) [0..] :: [Int]
%o -- _Reinhard Zumkeller_, Nov 11 2013, Dec 17 2011, Mar 28 2011
%o (PARI) for(n=0,90,v=digits(n);for(i=1,#v,print1(v[i]", "))) \\ _Charles R Greathouse IV_, Nov 20 2012
%o (PARI) apply( A007376(n)={for(k=1,n, k*10^k>n&& return(digits(n\k)[n%k+1]); n+=10^k)}, [0..200]) \\ _M. F. Hasler_, Nov 03 2019
%o (Magma) &cat[Reverse(IntegerToSequence(n)):n in[0..31]]; // _Jason Kimberley_, Dec 07 2012
%o (Python) A007376_list = [int(d) for n in range(10**2) for d in str(n)] # _Chai Wah Wu_, Feb 04 2015
%Y Considered as a sequence of digits, this is the same as the decimal expansion of the Champernowne constant, A033307. See that entry for a formula for a(n), further references, etc.
%Y Cf. A054632 (partial sums), A023103.
%Y For "decimations" see A127050 A127353 A127414 A127508 A127584 A127734 A127794 A127950 A128178 A128211 A128359 A128423 A128475 A128881.
%Y Cf. A193428, A256100, A001477 (the nonnegative integers), A117804, A100470.
%Y Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), this sequence and A033307 (b=10). - _Jason Kimberley_, Dec 06 2012
%Y Row lengths in A055642.
%Y For primes here see A071620. See A007908 for a very similar sequence.
%Y Cf. A058183.
%K base,easy,nice,nonn,tabf
%O 0,3
%A _N. J. A. Sloane_, _Robert G. Wilson v_
%E Extended to a(0) = 0 by _M. F. Hasler_, Oct 23 2019
|
