A276349 Numbers consisting of a nonempty string of 1's followed by a nonempty string of 0's.

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

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