A299872 -id:A299872 - cp's OEIS frontend

A300000 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, with a(1) = 1.

1, 10, 99, 999, 9990, 99900, 999000, 9990000, 99900000, 999000000, 9990000000, 99899999991, 998999999919, 9989999999190, 99899999991900, 998999999918991, 9989999999189910, 99899999991899109, 998999999918991090, 9989999999189910900, 99899999991899108991, 998999999918991089910, 9989999999189910899100

Offset: 1

Eric Angelini, Feb 10 2018

The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.
By definition, Sum_{k=1..n} a(k) = c(n) = concatenation of the first n digits of the sequence, therefore a(n) = c(n) - c(n-1). For n > 2, this defines a(n) recursively, without the need for solving an implicit equation, as the definition might suggest. - M. F. Hasler, Feb 22 2018
From Michael S. Branicky, Dec 07 2020: (Start)
Each digit 0-9 eventually appears.  The digit 1 appears first in a(1), 0 in a(2), 9 in a(3), 8 in a(12), 2 in a(68), 7 in a(71), 3 in a(2280), 6 in a(2283), 5 in a(2417), and 4 in a(4280).
All ten digits appear in each of a(4280) through a(121000).
Conjecture: a(n) contains all ten digits for n >= 4280. (End)
Conjecture holds through a(169000). - Michael S. Branicky, Jul 08 2022

			1 + 10 = 11 which is the concatenation of 1 and 1.
1 + 10 + 99 = 110 which is the concatenation of 1, 1 and 0.
1 + 10 + 99 + 999 = 1109 which is the concatenation of 1, 1, 0 and 9.
Otherwise said:
a(3) = concat(1,1,0) - (1 + 10) = 110 - 11 = 99,
a(4) = concat(1,1,0,9) - (11 + 99) = 1109 - 110 = 999,
a(5) = concat(1,1,0,9,9) - 1109 = 11099 - 1109 = 9990,
a(6) = concat(1,1,0,9,9,9) - 11099 = 99900, etc. - _M. F. Hasler_, Feb 22 2018

Michael S. Branicky, Table of n, a(n) for n = 1..1001 (terms 1..300 from Jean-Marc Falcoz)

A299865, A299866, A299867, A299868, A299869, A299870, A299871 and A299872 show the same type of sequence but with a different start.

The partial sums (the sequence c(n) mentioned in the Comments) is A299301.

a[1]=1;a[2]=10;a[n_]:=a[n]=FromDigits[Flatten[IntegerDigits/@Table[a[k],{k,n-1}]][[;;n]]]-Total@Table[a[m],{m,n-1}];
Table[a[l],{l,30}]  (* Giorgos Kalogeropoulos, May 20 2019 *)

a(n, show=1, a=1, c=a, d=[c])={for(n=2, n, show&&print1(a","); a=-c+c=c*10+d[1]; d=concat(d[^1],if(n>2,digits(a)))); a} \\ M. F. Hasler, Feb 22 2018

def a(n):
    alist, c, ckm1 = [1, 10], "110", 11
    for k in range(3, n+1):
        ck = 10*ckm1 + int(c[k-1])
        ak, ckm1 = ck - ckm1, ck
        c += str(ak)
        alist.append(ak)
    return alist[n-1]
print([a(n) for n in range(1, 24)]) # Michael S. Branicky, Dec 07 2020

a(n) = c(n) - c(n-1), where c(n) is the concatenation of the first n digits. c(n) ~ 1.1*10^(n-1), and a(n) ~ 0.999*10^(n-1). - M. F. Hasler, Feb 22 2018

A299865 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 2.

Original entry on oeis.org

2, 20, 198, 1981, 19818, 198179, 1981783, 19817838, 198178379, 1981783783, 19817837830, 198178378308, 1981783783079, 19817837830783, 198178378307837, 1981783783078363, 19817837830783638, 198178378307836379, 1981783783078363783, 19817837830783637836, 198178378307836378362, 1981783783078363783612

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Feb 21 2018

The sequence starts with a(1) = 2 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

			2 + 20 = 22 which is the concatenation of 2 and 2.
2 + 20 + 198 = 220 which is the concatenation of 2, 2 and 0.
2 + 20 + 198 + 1981 = 2201 which is the concatenation of 2, 2, 0 and 1.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..300

A300000 is the lexicographically first sequence of this type, with a(1) = 1.

Cf. A299866, ..., A299872 for variants with a(1) = 3, ..., 9.

a(n,show=1,a=2,c=a,d=[c])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018

a(n) = c(n) - c(n-1), where c(n) = concatenation of the first n digits, c(n) ~ 0.22*10^n, a(n) ~ 0.198*10^n. See A300000 for the proof. - M. F. Hasler, Feb 22 2018

A299866 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, a(1) = 3.

Original entry on oeis.org

3, 30, 297, 2972, 29727, 297268, 2972675, 29726757, 297267568, 2972675675, 29726756750, 297267567507, 2972675675068, 29726756750675, 297267567506755, 2972675675067545, 29726756750675457, 297267567506754568, 2972675675067545675, 29726756750675456754, 297267567506754567542

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Feb 21 2018

The sequence starts with a(1) = 3 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

			3 + 30 = 33 which is the concatenation of 3 and 3.
3 + 30 + 297 = 330 which is the concatenation of 3, 3 and 0.
3 + 30 + 297 + 2972 = 3302 which is the concatenation of 3, 3, 0 and 2.
From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 330 - 33 = 297, a(4) = 3302 - 330 = 2972, etc. - _M. F. Hasler_, Feb 22 2018

Jean-Marc Falcoz, Table of n, a(n) for n = 1..300

A300000 is the lexicographically first sequence of this type, with a(1) = 1.

Cf. A299865, ..., A299872 for variants with a(1) = 2, ..., 9.

a(n,show=1,a=3,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018

a(n) = c(n) - c(n-1), where c(n) = concatenation of the first n digits, c(n) ~ 0.33*10^n, a(n) ~ 0.297*10^n. See A300000 for the proof. - M. F. Hasler, Feb 22 2018

A299871 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 8.

Original entry on oeis.org

8, 80, 792, 7927, 79272, 792713, 7927135, 79271352, 792713513, 7927135135, 79271351350, 792713513502, 7927135135013, 79271351350135, 792713513501345, 7927135135013455, 79271351350134552, 792713513501345513, 7927135135013455135, 79271351350134551344, 792713513501345513442, 7927135135013455134424

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Feb 21 2018

The sequence starts with a(1) = 8 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

			8 + 80 = 88 which is the concatenation of 8 and 8.
8 + 80 + 792 = 880 which is the concatenation of 8, 8 and 0.
8 + 80 + 792 + 7927 = 8807 which is the concatenation of 8, 8, 0 and 7.
From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 880 - 88 = 792, a(4) = 8807 - 880 = 7927, etc. - _M. F. Hasler_, Feb 22 2018

Jean-Marc Falcoz, Table of n, a(n) for n = 1..300

A300000 is the lexicographically first sequence of this type, with a(1) = 1.

Cf. A299865, ..., A299872 for variants with a(1) = 2, ..., 9.

a(n,show=1,a=8,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018

a(n) = c(n) - c(n-1), where c(n) = concatenation of the first n digits, c(n) ~ 0.88*10^n, a(n) ~ 0.79*10^n. See A300000 for the proof. - M. F. Hasler, Feb 22 2018

A299867 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 4.

Original entry on oeis.org

4, 40, 396, 3963, 39636, 396357, 3963567, 39635676, 396356757, 3963567567, 39635675670, 396356756706, 3963567567057, 39635675670567, 396356756705673, 3963567567056727, 39635675670567276, 396356756705672757, 3963567567056727567, 39635675670567275672, 396356756705672756722

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Feb 21 2018

The sequence starts with a(1) = 4 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

			4 + 40 = 44 which is the concatenation of 4 and 4.
4 + 40 + 396 = 440 which is the concatenation of 4, 4, and 0.
4 + 40 + 396 + 3963 = 4403 which is the concatenation of 4, 4, 0 and 3.
From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 440 - 4 = 396, a(4) = 4403 - 440 = 3963, etc. - _M. F. Hasler_, Feb 22 2018

Jean-Marc Falcoz, Table of n, a(n) for n = 1..300

A300000 is the lexicographically first sequence of this type, with a(1) = 1.

Cf. A299865, ..., A299872 for variants with a(1) = 2, ..., 9.

a(n,show=1,a=4,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018

a(n) = c(n) - c(n-1), where c(n) = concatenation of the first n digits, c(n) ~ 0.44*10^n, a(n) ~ 0.396*10^n. See A300000 for the proof. - M. F. Hasler, Feb 22 2018

A299868 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 5.

Original entry on oeis.org

5, 50, 495, 4954, 49545, 495446, 4954459, 49544595, 495445946, 4954459459, 49544594590, 495445945905, 4954459459046, 49544594590459, 495445945904591, 4954459459045909, 49544594590459095, 495445945904590946, 4954459459045909459, 49544594590459094590, 495445945904590945902, 4954459459045909459018

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Feb 21 2018

The sequence starts with a(1) = 5 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

			5 + 50 = 55 which is the concatenation of 5 and 5.
5 + 50 + 495 = 550 which is the concatenation of 5, 5 and 0.
5 + 50 + 495 + 4954 = 5504 which is the concatenation of 5, 5, 0 and 4.
From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 550 - 55 = 495, a(4) = 5504 - 550 = 4954, etc. - _M. F. Hasler_, Feb 22 2018

Jean-Marc Falcoz, Table of n, a(n) for n = 1..300

A300000 is the lexicographically first sequence of this type, with a(1) = 1.

Cf. A299865, ..., A299872 for variants with a(1) = 2, ..., 9.

a(n,show=1,a=5,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018

a(n) = c(n) - c(n-1), where c(n) = concatenation of the first n digits, c(n) ~ 0.55*10^n, a(n) ~ 0.495*10^n. See A300000 for the proof. - M. F. Hasler, Feb 22 2018

A299869 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 6.

Original entry on oeis.org

6, 60, 594, 5945, 59454, 594535, 5945351, 59453514, 594535135, 5945351351, 59453513510, 594535135104, 5945351351035, 59453513510351, 594535135103509, 5945351351035091, 59453513510350914, 594535135103509135, 5945351351035091351, 59453513510350913508, 594535135103509135082, 5945351351035091350820

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Feb 21 2018

The sequence starts with a(1) = 6 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

			6 + 60 = 66 which is the concatenation of 6 and 6.
6 + 60 + 594 = 660 which is the concatenation of 6, 6 and 0.
6 + 60 + 594 + 5945 = 6605 which is the concatenation of 6, 6, 0 and 5.
From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 660 - 66 = 594, a(4) = 6605 - 660 = 5945, etc. - _M. F. Hasler_, Feb 22 2018

Jean-Marc Falcoz, Table of n, a(n) for n = 1..300

A300000 is the lexicographically first sequence of this type, with a(1) = 1.

Cf. A299865, ..., A299872 for variants with a(1) = 2, ..., 9.

a(n,show=1,a=6,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018

a(n) = c(n) - c(n-1), where c(n) = concatenation of the first n digits, c(n) ~ 0.66*10^n, a(n) ~ 0.59*10^n. See A300000 for the proof. - M. F. Hasler, Feb 22 2018

A299870 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 7.

Original entry on oeis.org

7, 70, 693, 6936, 69363, 693624, 6936243, 69362433, 693624324, 6936243243, 69362432430, 693624324303, 6936243243024, 69362432430243, 693624324302427, 6936243243024273, 69362432430242733, 693624324302427324, 6936243243024273243, 69362432430242732426, 693624324302427324262, 6936243243024273242622

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Feb 21 2018

The sequence starts with a(1) = 7 and is always extended with the smallest integer not yet present in the sequence and not leading to a contradiction.

			7 + 70 = 77 which is the concatenation of 7 and 7.
7 + 70 + 693 = 770 which is the concatenation of 7, 7 and 0.
7 + 70 + 693 + 6936 = 7706 which is the concatenation of 7, 7, 0 and 6.
From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 770 - 77 = 693, a(4) = 7706 - 770 = 6936, etc. - _M. F. Hasler_, Feb 22 2018

Jean-Marc Falcoz, Table of n, a(n) for n = 1..300

A300000 is the lexicographically first sequence of this type, with a(1) = 1.

Cf. A299865, ..., A299872 for variants with a(1) = 2, ..., 9.

a(n,show=1,a=7,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018

a(n) = c(n) - c(n-1), where c(n) = concatenation of the first n digits, c(n) ~ 0.77*10^n, a(n) ~ 0.69*10^n. See A300000 for the proof. - M. F. Hasler, Feb 22 2018

cp's OEIS Frontend

A300000 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, with a(1) = 1.

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A299865 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 2.

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A299866 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, a(1) = 3.

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A299871 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 8.

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A299867 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 4.

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A299868 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 5.

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A299869 The sum of the first n terms of the sequence is the concatenation of the first n digits of the sequence, and a(1) = 6.

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