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A276349 Numbers consisting of a nonempty string of 1's followed by a nonempty string of 0's. 3
10, 100, 110, 1000, 1100, 1110, 10000, 11000, 11100, 11110, 100000, 110000, 111000, 111100, 111110, 1000000, 1100000, 1110000, 1111000, 1111100, 1111110, 10000000, 11000000, 11100000, 11110000, 11111000, 11111100, 11111110, 100000000, 110000000, 111000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Intersection of A037415 and A009996 except for 1 [Corrected by David A. Corneth, Aug 30 2016].

Set of terms from sequence A052983.

a(n) is the binary expansion of A043569(n). - Michel Marcus, Sep 04 2016

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Luboš Pick, Dirichletovy šuplíčky, Pokroky matematiky, fyziky a astronomie, Vol. 61, No. 2 (2016), pp. 106-118. (In Czech; The Dirichlet pigeonhole principle)

FORMULA

A227362(a(n)) = 10.

From Robert Israel, Sep 02 2016: (Start)

a((m^2-m)/2+j) = 10^(m+1)*(1-10^(-j))/9 for m>=1, 1<=j<=m.

a(n) = 10*(10^m - 10^(-n+m*(m+1)/2))/9 where m = A002024(n). (End)

A002275(A002260(n)) * 10^A004736(n) - Peter Kagey, Sep 02 2016

Sum_{n>=1} 1/a(n) = A073668. - Amiram Eldar, Feb 20 2022

EXAMPLE

60 is of the form binomial(a, 2) + b where 0 < b <= a and a = 11, b = 5. So a(60) has (11 + 1) digits and 5 leading ones. The other digits are 0. Giving a(60) = 111110000000. It has 7 (more than 1) trailing zeros so the next one, a(61) is a(60) + 10^(7 - 1). - David A. Corneth, Aug 30 2016

MAPLE

seq(seq(10^(m+1)*(1-10^(-j))/9, j=1..m), m=1..20); # Robert Israel, Sep 02 2016

MATHEMATICA

Table[FromDigits@ Join[ConstantArray[1, #1], ConstantArray[0, #2]] & @@@ Transpose@ {#, n - #} &@ Range[n - 1], {n, 2, 9}] // Flatten (* Michael De Vlieger, Aug 30 2016 *)

Flatten[Table[FromDigits[Join[PadRight[{}, n, 1], PadRight[{}, k, 0]]], {n, 8}, {k, 8}]]//Sort (* Harvey P. Dale, Jan 09 2019 *)

PROG

(Magma) [n: n in [1..10^7] | Seqint(Setseq(Set(Sort(Intseq(n))))) eq 10 and Seqint(Sort((Intseq(n)))) eq n]

(PARI) is(n) = vecmin(digits(n))==0 && vecmax(digits(n))==1 && digits(n)==vecsort(digits(n), , 4) \\ Felix Fröhlich, Aug 30 2016

(PARI) a(n) = my(r = ceil((sqrt(1+8*n)+1)/2), k = n - binomial(r-1, 2)); 10^(r-k)*(10^(k)-1)/9

\\ given an element n, computes the next element of the sequence.

nxt(n) = my(d = digits(n), qd=#d, s = vecsum(d)); if(qd-s>1, n+10^(qd-s-1), 10^qd)

\\ given an element n of the sequence, computes its place in the sequence.

inv(n) = my(d = digits(n)); binomial(#d-1, 2) + vecsum(d) \\ David A. Corneth, Aug 31 2016

CROSSREFS

Cf. A002024, A009996, A037415, A043569, A052983, A073668, A227362, A276348.

Sequence in context: A154810 A099820 A273245 * A167502 A135652 A035504

Adjacent sequences: A276346 A276347 A276348 * A276350 A276351 A276352

KEYWORD

nonn,base

AUTHOR

Jaroslav Krizek, Aug 30 2016

STATUS

approved

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Last modified November 26 13:43 EST 2022. Contains 358362 sequences. (Running on oeis4.)