A186074 -id:A186074 - 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)

A070152 Take pairs (x,y) with Sum_{j = x..y} j = concatenation of x and y. Sort pairs on y then x. This sequence gives x of each pair.

Original entry on oeis.org

1, 2, 4, 13, 18, 33, 35, 7, 78, 133, 178, 228, 273, 388, 710, 1333, 1701, 1778, 2737, 3273, 3563, 3087, 3478, 12488, 13333, 14208, 17778, 31463, 36993, 5338, 7063, 9063, 12643, 15238, 17147, 22448, 23788, 27313, 29058, 34488, 36763, 38788, 43273, 50813, 53578

Offset: 1

Lekraj Beedassy, May 06 2002

From Bernard Schott, Jan 26 2022: (Start)
Some subsequences, from Diophante and Crux Mathematicorum:
{(2*10^m-5)/15, m >= 1} = 1, 13, 133, 1333, ... = A097166.
{2*(4*10^m+5)/45, m >= 1} = 2, 18, 178, 1778, ...
{13*(26*100^m-125)/12375, m >= 2} = 273, 27313, 2731313, ... (End)

			1+...+5 = 15; 2+...+7 = 27; 4+...+29 = 429; 13+...+53 = 1353; 18+...+63 = 1863.
133+...+533 = 133533.
273+...+2353 = 2732353.

Diophante, A1945 - Concaténations en tous genres (in French).
R. Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum 27(1), 2001, p. 39-44.

Cf. A070153, A071297, A186074.

Subsequence: A097166.

More terms from David W. Wilson, Jun 04 2002

Name edited by Michel Marcus, Jan 29 2022

A070153 Take pairs (x,y) with Sum_{j = x..y} j = concatenation of x and y. Sort pairs on y then x. This sequence gives y of each pair.

Original entry on oeis.org

5, 7, 29, 53, 63, 88, 91, 119, 403, 533, 623, 2148, 2353, 2813, 3835, 5333, 6076, 6223, 7889, 8728, 9163, 25039, 26603, 51513, 53333, 55168, 62223, 85338, 93633, 103463, 119063, 134938, 159518, 175238, 185979, 213073, 219413, 235313, 242818, 264888

Offset: 1

Lekraj Beedassy, May 06 2002

From Bernard Schott, Jan 26 2022: (Start)
Some subsequences, from Diophante and Crux Mathematicorum:
{(8*10^m-5)/15, m >= 1} = 5, 53, 533, 5333, ... (A350995).
{7*(4*10^m+5)/45, m >= 1} = 7, 63, 623, 6223, ...
{13*(224*100^m-125)/12375, m >= 2} = 2353, 235313, 23531313, ... (End)

			1+...+5 = 15; 2+...+7 = 27; 4+...+29 = 429; 13+...+53 = 1353; 18+...+63 = 1863.
133+...+533 = 133533.
178+...+623 = 178623.

Diophante, A1945 - Concaténations en tous genres (in French).
R. Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum 27(1), 2001, p. 39-44.

Cf. A070152, A071297, A097166, A186074.

Cf. A350995 (is a subsequence).

More terms from David W. Wilson, Jun 04 2002

Name edited by Michel Marcus, Jan 29 2022

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)

A350994 a(n) = (40100^n + 610^n - 1)/3.

Original entry on oeis.org

15, 1353, 133533, 13335333, 1333353333, 133333533333, 13333335333333, 1333333353333333, 133333333533333333, 13333333335333333333, 1333333333353333333333, 133333333333533333333333, 13333333333335333333333333, 1333333333333353333333333333, 133333333333333533333333333333

Offset: 0

Bernard Schott, Jan 28 2022

Subsequence of A186074.

Terms of this sequence satisfy the identity proposed in 2nd formula because a(n) = Sum_{j=(4*10^n-1)/3..(16*10^n-1)/3} j = ((4*10^n-1)/3).((16*10^n-1)/3) where "." means concatenation (see examples).

			a(0) = (40+6-1)/3 = Sum_{j=1..5} j = 15.
a(1) = (4000+60-1)/3 = Sum_{j=13..53} j = 1353.
a(2) = (400000+600-1)/3 = Sum_{j=133..533} j = 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 (111,-1110,1000).

Cf. A070152, A070153, A097166, A186074, A332167, A350995.

Data := seq((40*100^n + 6*10^n - 1)/3,  n = 0..17);

Table[(40*100^n + 6*10^n - 1)/3, {n, 0, 17}] (* Amiram Eldar, Jan 29 2022 *)
LinearRecurrence[{111,-1110,1000},{15,1353,133533},20] (* Harvey P. Dale, Jun 16 2025 *)

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3), n >= 3.
a(n) = Sum_{j=A097166(n)..A350995(n)} j = A097166(n).A350995(n) where "." means concatenation.
G.f.: (15 - 312*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Stefano Spezia, Jan 30 2022
a(n) = 2*A332167(n) + 1. - Hugo Pfoertner, Jan 30 2022
E.g.f.: exp(x)*(40*exp(99*x) + 6*exp(9*x) - 1)/3. - Elmo R. Oliveira, May 02 2025

A186076 Numbers m such that m = Sum_{i=x..y} i = (10^k)*y + x, where 0 <= x < y, 0 <= x < 10^k for some positive integers k.

Original entry on oeis.org

190, 204, 216, 19900, 20328, 21252, 21762, 23287, 23490, 1999000, 2002077, 2006118, 2077402, 2132532, 2177622, 199990000, 202272147, 202722352, 203872812, 207093834, 213325332, 217006075, 217776222, 227367888, 232728727, 235629162, 19999900000, 20001201612

Offset: 1

Matthew Goers, Feb 11 2011

Numbers that are the sum from their more significant digits counted down to the following digits. The first is the 19th triangular number: 190 = 19 + 18 + 17 + ... + 1 + 0.
See A186074 for numbers that sum by counting upward.
An alternate definition: n = sum from x to y inclusive = A000217(y) - A000217(x-1), (A000217 are the triangular numbers) where the digits of n are the concatenation of y and x.
These are the positive integer solutions to the equation Sum_{i=x..y} i = (10^k)*y + x, where 0 <= x < y, 0 <= x < 10^k, k = 1,2,3...
The graph of the function is a hyperbola; the solutions are for positive x and y, where x does not "overlap" and add to y. The first 15 terms are all of the solutions for m = 1 to 3.
Note that terms A186074(4) and A186074(10) have trailing 0's, i.e. 19900 = Sum_{k=0..199} k and 1999000 = Sum_{k=0..1999} k. Strictly speaking, these do not meet the concatenation criterion. This pattern continues indefinitely: 199990000, 19999900000, etc. - Matthew Goers, Jun 03 2011
All terms form (10^k)*y + x, where y = (s+t-1)/2 + 10^k, x = (s-t-1)/2, s*t = 100^k - 10^k, 0 <= (s-t-1)/2 < 10^k, and gcd(s, t) is an odd number. - Jinyuan Wang, Sep 13 2019

			204 = 20 + 19 + 18 + ... + 5 + 4.
2002077 = 2002 + 2001 + ... + 78 + 77.
2006118 = 2006 + 2005 + ... + 119 + 118.

Jinyuan Wang, Table of n, a(n) for n = 1..21167

Cf. A186074, A000217.

do(s, t, k) = if(s - t > 0 && (s-t-1)/2 < 10^k, (10^k-1+s)*(10^k+1+t)/2, 204);
lista(nn) = {my(v=List()); for(k = 1, nn, fordiv(50^k - 5^k, s, t = (100^k-10^k)/s; listput(v, do(s, t, k)); listput(v, do(2^k*s, t/2^k, k)))); Set(v); } \\ Jinyuan Wang, Sep 13 2019

Missing term a(4) = 19900 inserted by Matthew Goers, Jun 03 2011

a(16)-a(28) from Donovan Johnson, Aug 22 2012

cp's OEIS Frontend

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

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A070152 Take pairs (x,y) with Sum_{j = x..y} j = concatenation of x and y. Sort pairs on y then x. This sequence gives x of each pair.

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A070153 Take pairs (x,y) with Sum_{j = x..y} j = concatenation of x and y. Sort pairs on y then x. This sequence gives y of each pair.

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

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A350994 a(n) = (40*100^n + 6*10^n - 1)/3.

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Maple

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Formula

A186076 Numbers m such that m = Sum_{i=x..y} i = (10^k)*y + x, where 0 <= x < y, 0 <= x < 10^k for some positive integers k.

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

A350994 a(n) = (40100^n + 610^n - 1)/3.