A350994 -id:A350994 - cp's OEIS frontend

A097166 Expansion of g.f. (1+2x)/((1-x)(1-10*x)).

1, 13, 133, 1333, 13333, 133333, 1333333, 13333333, 133333333, 1333333333, 13333333333, 133333333333, 1333333333333, 13333333333333, 133333333333333, 1333333333333333, 13333333333333333, 133333333333333333, 1333333333333333333, 13333333333333333333, 133333333333333333333

Offset: 0

Paul Barry, Jul 30 2004

Partial sums of (1+2*x)/(1-10*x) = {1, 12, 120, 1200, ...}.
Second binomial transform of A082365.
These terms are the x's of A070152 and the corresponding y's are A350995 (see formula and examples). - Bernard Schott, Feb 15 2022

			a(0) = (4-1)/3 = 1 and Sum_{j=1..5} = 15.
a(1) = (40-1)/3 = 13 and Sum_{j=13..53} = 1353.
a(2) = (400-1)/3 = 133 and Sum_{j=133..533} = 133533.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Diophante, A1945 - Concaténations en tous genres (in French).
Richard Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum with Mathematical Mayhem 27:1 (2001), pp. 39-44.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A056698 (index of primes), A082365, A097169, A309907 (squares of this).

Cf. A070152, A070153, A186074, A198970, A350994, A350995.

[(4*10^n-1)/3 : n in [0..20]]; // Vincenzo Librandi, Nov 01 2011

a:= n-> parse(cat(1, 3$n)):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 23 2019

NestList[10#+3&,1,20] (* Harvey P. Dale, Jan 22 2014 *)

[(4*10**n-1)//3 for n in range(25)] # Gennady Eremin, Mar 04 2022

a(n) = (4*10^n - 1)/3.
a(n) = A097169(2*n).
a(n) = 10*a(n-1) + 3, n>0. a(n) = 11*a(n-1) - 10*a(n-2), n>1. - Vincenzo Librandi, Nov 01 2011
A350994(n) = Sum_{j=a(n)..A350995(n)} = a(n).A350995(n) where "." means concatenation. - Bernard Schott, Jan 28 2022
From Elmo R. Oliveira, Apr 29 2025: (Start)
E.g.f.: exp(x)*(4*exp(9*x) - 1)/3.
a(n) = A198970(n)/3. (End)

A186074 Numbers k such that k = Sum_{i=x..y} i and, in decimal, k is the concatenation of x and y.

Original entry on oeis.org

15, 27, 429, 1353, 1863, 3388, 3591, 7119, 78403, 133533, 178623, 2282148, 2732353, 3882813, 7103835, 13335333, 17016076, 17786223, 27377889, 32738728, 35639163, 308725039, 347826603, 1248851513, 1333353333, 1420855168, 1777862223, 3146385338, 3699393633

Offset: 1

Matthew Goers, Feb 11 2011

The sum from one set of digits to the following set of digits equals the term. The first is the 5th triangular number: 15 = 1 + 2 + 3 + 4 + 5.
These are the positive integer solutions for the formula sum (x to y) = (10^k)*x + y, where 0 < x < y < 10^k for some k >= 1.
On the left hand side of this equation, the sum can be written as A000217(y) - A000217(x-1) = (x+y)*(1-x+y)/2, and the right hand side is the concatenation of the decimal digits of x and y.
The graph of the function is a hyperbola; the solutions are for positive x and y, where y does not "overlap" and add to x. The first 21 terms are all of the solutions for n = 1 to 4. n = 5 solutions add two 9-digit and six 10-digit terms.
Note the pattern 15 = sum (1 to 5); 1353 = sum (13 to 53); 133533 = sum (133 to 533); 13335333 = sum (1333 to 5333). This pattern continues: 1333353333 = sum (13333 to 53333); 133333533333 = sum (133333 to 533333); etc. These are not the next terms in the sequence, however (see A350994).
See A186076 for the case of a sum countdown from the more significant to less significant digits.
From Jinyuan Wang, Sep 14 2019: (Start)
All terms form the concatenation of x = (s+t+1)/2 - 10^k and y = (s-t+1)/2, where s*t = 100^k - 10^k, 10^(k-1) < (s-t+1)/2 < 10^k, and gcd(s, t) is an odd number.
Strictly speaking, 1301613 = 13 + 14 + ... + 1612 + 1613 does not meet the concatenation criterion. So 1301613 is not a term.
(End)
From Bernard Schott, Jan 26 2022: (Start)
Note that numbers x are in A070152, and corresponding y in A070153 (see formula).
Other subsequence pattern: 27, 1863, 178623, 17786223, 1777862223, ... where 17..78 +...+ 62..23 = 17..7862..23. (End)

			429 = 4 + 5 + 6 + ... + 28 + 29.
7119 = 7 + 8 + 9 + ... + 118 + 119.
3882813 = 388 + 389 + ... + 2812 + 2813.

Jinyuan Wang, Table of n, a(n) for n = 1..18738 (first 142 terms from David A. Corneth)
Diophante, A1945 - Concaténations en tous genres (in French).
Richard Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum with Mathematical Mayhem 27:1 (2001), pp. 39-44.

Cf. A186076.
Cf. A070152, A070153.
Cf. A350994 (subsequence).

# See "Astonishing Pairs of Numbers" article referenced above.

do(s, t, k) = if(10^(k-1) < (s-t+1)/2 && (s-t+1)/2 < 10^k, (1-10^k+s)*(1+10^k-t)/2);
lista(nn) = {my(m, v=List()); for(k = 1, nn, fordiv(50^k - 5^k, s, t = (100^k-10^k)/s; if(m=do(s, t, k), listput(v, m)); if(m=do(2^k*s, t/2^k, k), listput(v, m)))); vecsort(Vec(v)); } \\ Jinyuan Wang, Aug 29 2019

a(n) = A070152(n).A070153(n) where "." means concatenation. - Bernard Schott, Jan 29 2022

a(22)-a(29) from Matthew Goers, Apr 11 2013

A350995 a(n) = (16*10^n - 1)/3.

Original entry on oeis.org

5, 53, 533, 5333, 53333, 533333, 5333333, 53333333, 533333333, 5333333333, 53333333333, 533333333333, 5333333333333, 53333333333333, 533333333333333, 5333333333333333, 53333333333333333, 533333333333333333, 5333333333333333333, 53333333333333333333, 533333333333333333333, 5333333333333333333333

Offset: 0

Bernard Schott, Jan 28 2022

These terms 'y' form a subsequence of A070153 and the corresponding terms 'x' are in A097166 (see 3rd formula and examples).

			a(0) = (16-1)/3 = 5 and Sum_{j=1..5} = 15.
a(1) = (160-1)/3 = 53 and Sum_{j=13..53} = 1353.
a(2) = (1600-1)/3 = 533 and Sum_{j=133..533} = 133533.

Diophante, A1945 - Concaténations en tous genres (in French).
Richard Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum with Mathematical Mayhem 27:1 (2001), pp. 39-44.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Subsequence of A070153.

Cf. A070152, A097166, A186074, A350994.

Maple
```
Data := seq((16*10^n-1)/3,  n = 0..21);
```

Table[(16*10^n - 1)/3, {n, 0, 21}] (* Amiram Eldar, Jan 29 2022 *)

a(n) = 10*a(n-1) + 3, n>0.
a(n) = 11*a(n-1) - 10*a(n-2), n>1.
A350994(n) = Sum_{j=A097166(n)..a(n)} = A097166(n).a(n) where "." means concatenation.
From Elmo R. Oliveira, May 02 2025: (Start)
G.f.: (5-2*x)/((1-x)*(1-10*x)).
E.g.f.: exp(x)*(16*exp(9*x) - 1)/3. (End)

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A097166 Expansion of g.f. (1+2x)/((1-x)(1-10*x)).

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A186074 Numbers k such that k = Sum_{i=x..y} i and, in decimal, k is the concatenation of x and y.

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A350995 a(n) = (16*10^n - 1)/3.

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cp's OEIS Frontend

A097166 Expansion of g.f. (1+2*x)/((1-x)*(1-10*x)).

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Magma

Maple

Mathematica

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Formula

A186074 Numbers k such that k = Sum_{i=x..y} i and, in decimal, k is the concatenation of x and y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A350995 a(n) = (16*10^n - 1)/3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A097166 Expansion of g.f. (1+2x)/((1-x)(1-10*x)).