A014925 -id:A014925 - cp's OEIS frontend

A104759 Concatenation of digits of natural numbers from n down to 1.

1, 21, 321, 4321, 54321, 654321, 7654321, 87654321, 987654321, 1987654321, 1987654321, 101987654321, 1101987654321, 11101987654321, 211101987654321, 1211101987654321, 31211101987654321, 131211101987654321, 4131211101987654321, 14131211101987654321, 514131211101987654321

Offset: 1

Alexandre Wajnberg & Juliette Bruyndonckx, Apr 23 2005

			a(11) = a(10) because no number may begin with 0.
a(9)= [123456789]101112131415...=987654321
a(10)=[1234567891]01112131415...=1987654321
a(11)=[12345678910]1112131415...=01987654321=1987654321
a(12)=[123456789101]112131415...=101987654321
a(13)=[1234567891011]12131415...=1101987654321
a(14)=[12345678910111]2131415...=11101987654321
a(15)=[123456789101112]131415...=211101987654321

Michael De Vlieger, Table of n, a(n) for n = 1..1000 (last term has 1000 decimal digits)
Eric Weisstein's World of Mathematics, Consecutive numbers sequences.

Cf. A014925, A000422, A057138, A060554, A138793.

f[n_] := Block[{t = Reverse@ Flatten@ IntegerDigits@ Range@ n, k}, Reap@ For[k = 1, k <= Length@ t, k++, Sow[FromDigits@ Take[t, -k]]] // Flatten // Rest]; f@ 14 (* Michael De Vlieger, Mar 23 2015 *)
lst = {}; Do[lst = Join[lst, IntegerDigits[n]], {n, 1, 100}]; Table[FromDigits[Reverse[lst[[Range[1, n]]]]], {n, 1, Length[lst]}] (* Robert Price, Mar 24 2015 *)

a(n) = A138793(n) mod 10^(n-1). - R. J. Mathar, Sep 17 2011

A272525 Convolution of nonzero repunits (A002275) with themselves.

Original entry on oeis.org

1, 22, 343, 4664, 58985, 713306, 8367627, 96021948, 1083676269, 12071330590, 133058984911, 1454046639232, 15775034293553, 170096021947874, 1824417009602195, 19478737997256516, 207133058984910837, 2194787379972565158, 23182441700960219479, 244170096021947873800

Offset: 0

Ilya Gutkovskiy, May 02 2016

Partial sums of A014925.

Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (22,-141,220,-100)

Cf. A002275, A010879, A014925, A083449.

LinearRecurrence[{22, -141, 220, -100}, {1, 22, 343, 4664}, 20]
Table[(9 n (10^(n + 2) + 1) + 7 10^(n + 2) + 29)/729, {n, 0, 19}]

A272525(n)=(9*n+7)*(10^(n+2)+1)\729+1 \\ M. F. Hasler, Nov 02 2016

O.g.f.: 1/((1 - 10*x)^2*(1 - x)^2).
E.g.f.: (29 + 9*x  + 700*exp(9*x) + 9000*x*exp(9*x))*exp(x)/729.
a(n) =  22*a(n-1) - 141*a(n-2) + 220*a(n-3) - 100*a(n-4).
a(n) = (9*n(10^(n+2) + 1) + 7*10^(n+2) + 29)/729.
A010879(a(n)) = A010879(n+1).

A317824 a(n) = A000422(n)^^A000422(n) (mod 10^len(A000422(n))), where ^^ indicates tetration or hyper-4 (e.g., 3^^4 = 3^(3^(3^3))).

Original entry on oeis.org

1, 21, 721, 8721, 8721, 708721, 5708721, 65708721, 165708721, 65165708721, 1165165708721, 861165165708721, 5861165165708721, 5005861165165708721, 55005861165165708721, 48055005861165165708721, 8448055005861165165708721, 388448055005861165165708721, 49388448055005861165165708721

Offset: 1

Marco Ripà, Aug 10 2018

For any n, a(n) (mod 10^len(A000422(n))) == a(n + 1) (mod 10^len(A000422(n))), where len(k) := number of digits in k. Assuming len(a(n))>1, this is a general property of every concatenated sequence with fixed rightmost digits (such as A061839 or A014925), as shown in Ripà's book "La strana coda della serie n^n^...^n".

			For n = 3, a(3) = 321^^321 (mod 10^3) = 721. In fact, a(3) (mod 10^3) == a(4) (mod 10^3), since 721 (mod 10^3) == 8721 (mod 10^3).

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. ISBN 978-88-6178-789-6

Marco Ripà, On the Convergence Speed of Tetration, ResearchGate (2018).
Wikipedia, Tetration

Cf. A000422, A058183, A171882 (tetration), A317903.

tmod(b, n) = {if (b % n == 0, return (0)); if (b % n == 1, return (1)); if (gcd(b, n)==1, return (lift(Mod(b, n)^tmod(b, lift(znorder(Mod(b, n))))))); lift(Mod(b, n)^(eulerphi(n) + tmod(b, eulerphi(n))));}
f(n) = my(t=n); forstep(k=n-1, 1, -1, t=t*10^#Str(k)+k); t; \\ A000422
a(n) = my(x=f(n)); tmod(x, 10^#Str(x)); \\ Michel Marcus, Sep 12 2021

a(n) = (n_n-1_n-2_...2_1)^^(n_n-1_n-2...2_1) (mod 10^len(n_n-1_n-2..._2_1)), where len(k) := number of digits in k.

More terms from Jinyuan Wang, Aug 30 2020

A317903 a(n) = A038394(n)^^A038394(n) (mod 10^len(A038394(n))), where ^^ indicates tetration or hyper-4 (e.g., 3^^4=3^(3^(3^3))).

Original entry on oeis.org

4, 76, 176, 4176, 314176, 91314176, 891314176, 80891314176, 88080891314176, 5288080891314176, 705288080891314176, 10705288080891314176, 2410705288080891314176, 912410705288080891314176, 42912410705288080891314176, 9242912410705288080891314176, 989242912410705288080891314176

Offset: 1

Marco Ripà, Aug 10 2018

For any n >= 2, a(n) (mod 10^len(A038394(n))) == a(n + 1) (mod 10^len(A038394(n))), where len(k) := number of digits in k. Assuming len(a(n))>1, this is a general property of every concatenated sequence with fixed rightmost digits (such as A014925 or A092447), as shown in Ripà's book "La strana coda della serie n^n^...^n".

			For n = 6, a(6) = 13117532^^13117532 (mod 10^8) == 91314176.

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. ISBN 978-88-6178-789-6

Marco Ripà, On the Convergence Speed of Tetration, ResearchGate (2018).

Cf. A038394, A068670, A171882 (tetration), A317824.

tmod(b, n) = {if (b % n == 0, return (0)); if (b % n == 1, return (1)); if (gcd(b, n)==1, return (lift(Mod(b, n)^tmod(b, lift(znorder(Mod(b, n))))))); lift(Mod(b, n)^(eulerphi(n) + tmod(b, eulerphi(n))));}
f(n) = fromdigits(concat([digits(p) | p<-Vecrev(primes(n))])); \\ A038394
a(n) = if (n==1, 4, my(x=f(n)); tmod(x, 10^#Str(x))); \\ Michel Marcus, Sep 12 2021

a(n) = (p(n)p(n-1)_p(n-2)...3_2)^^(p(n)_p(n-1)_p(n-2)...3_2) (mod 10^len(p(n)_p(n-1)_p(n-2)..._3_2)), where len(k) := number of digits in k.

More terms from Jinyuan Wang, Aug 30 2020

A057138 Add (n mod 10)*10^(n-1) to the previous term, with a(0) = 0.

Original entry on oeis.org

0, 1, 21, 321, 4321, 54321, 654321, 7654321, 87654321, 987654321, 987654321, 10987654321, 210987654321, 3210987654321, 43210987654321, 543210987654321, 6543210987654321, 76543210987654321, 876543210987654321

Offset: 0

Henry Bottomley, Aug 12 2000

Original definition: "Concatenate next digit at left hand end."
This is misleading, since the concatenation of 0 yields the same term (leading zeros vanish), but upon the next concatenation of 1, the 0 reappears - except for a(1), which according to that description should equal a(1)=10: It is surprising that in this only case where the 0 is indeed present, it disappears upon left-concatenation of the digit 1! - M. F. Hasler, Jan 13 2013
From Hieronymus Fischer, Jan 23 2013: (Start)
A definition which is also consistent is: Start with terms 0 and 1 and then concatenate the next digit at the left hand end. If the next digit is a zero, keep this zero in mind so that the following digit is a 1 preceding a 0.
The sequence terms are the terms of A057137 in reversed digit order. Based on this understanding, the anomaly for the indices 0 and 1 where the terms are 0 and 1 instead of 0 and 10 (what one would expect) becomes self-explaining. Also, the special behavior when the zero digit is encountered becomes clear.
Examples: a(3) = 321 = Reversal(A057137(3)),
a(10) = 987654321 = Reversal(A057137(10)) = Reversal(1234567890). (End)

Hieronymus Fischer, Table of n, a(n) for n = 0..200

Alternative progression for n >= 10 compared with A000422 and A014925.

Cf. A057137 for reverse.

ListTools:-PartialSums([seq((k mod 10)*10^(k-1), k=0..40)]); # Robert Israel, Jun 21 2017

Join[{c = 0}, Table[c = c + Mod[n, 10]*10^(n - 1), {n, 18}]] (* T. D. Noe, Jan 30 2013 *)
nxt[{n_,a_}]:={n+1,a+Mod[n+1,10]10^n}; NestList[nxt,{0,0},20][[;;,2]] (* Harvey P. Dale, Apr 06 2025 *)

a(n)=sum(i=0,n,i%10*10^(i-1)) \\ M. F. Hasler, Jan 13 2013

a(n) = a(n-1) + 10^(n-1)*n - 10^n*floor(n/10) = A057139(n) mod 10^n.
a(n) = floor(((q/(10^10 - 1)) + q mod 10^(n mod 10))*10^(10*floor(n/10))), where q = 987654321. - Hieronymus Fischer, Jan 03 2013
G.f.: x(1-10(10x)^9 + 9(10x)^10)/((1-x) (1-10x)^2 (1-(10x)^10)). - Robert Israel, Jun 21 2017

Better definition from M. F. Hasler, Jan 13 2013

A137233 Number of n-digit even numbers.

Original entry on oeis.org

5, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000

Offset: 1

Ctibor O. Zizka, Mar 08 2008

From Kival Ngaokrajang, Oct 18 2013: (Start)
a(n) is also the total number of double rows identified numbers in n digit.
For example:
  n = 1: 01 23 45 67 89 = 5 double rows;
  n = 2: 1011 1213 1415 1617 1819...9899 = 45 double rows;
  n = 3: 100101 102103 104105...998999 = 450 double rows;
The number of double rows is also A030656. (End)
a(n) is also the number of n-digit integers with an even number of even digits (A356929); a(5) = 45000 is the answer to the question 2 of the Olympiade Mathématique Belge in 2004 (link). - Bernard Schott, Sep 06 2022
a(n) is also the number of n-digit integers with an odd number of odd digits (A054684). - Bernard Schott, Nov 07 2022

			a(2) = 45 because there are 45 2-digit even numbers.

Olympiade Mathématique Belge, OMB 2004, Finale Maxi, Question 2.
Index to sequences related to Olympiads.
Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A000422, A002275, A002276, A011577, A014923, A014925, A016313, A019518, A037487, A053052, A057138, A090843, A097166, A099914, A099915.

Cf. A030656, A052268, A054684, A356929.

Vec(5*x*(1-x)/(1-10*x) + O(x^22)) \\ Elmo R. Oliveira, Jul 23 2025

def A137233(n): return 9*10**(n-1)+1>>1 # Chai Wah Wu, Nov 11 2022

a(n) = 9*10^(n-1)/2 if n > 1. - R. J. Mathar, May 23 2008
From Elmo R. Oliveira, Jul 23 2025: (Start)
G.f.: 5*x*(1-x)/(1-10*x).
E.g.f.: (-9 + 10*x + 9*exp(10*x))/20.
a(n) = 10*a(n-1) for n > 2.
a(n) = A052268(n)/2 for n >= 2. (End)

Corrected and extended by R. J. Mathar, May 23 2008

More terms from Elmo R. Oliveira, Jul 23 2025

A064616 a(n) = (10^n-1)(91/81)-n10^n/9.

Original entry on oeis.org

9, 89, 789, 6789, 56789, 456789, 3456789, 23456789, 123456789, 123456789, -9876543211, -209876543211, -3209876543211, -43209876543211, -543209876543211, -6543209876543211, -76543209876543211, -876543209876543211, -9876543209876543211

Offset: 1

Henry Bottomley, Sep 26 2001

Harry J. Smith, Table of n, a(n) for n=1..150
Index entries for linear recurrences with constant coefficients, signature (21,-120,100).

Cf. A002275, A014925, A064617.

LinearRecurrence[{21,-120,100},{9,89,789},20] (* Harvey P. Dale, May 15 2022 *)

{ a=0; q=1; for (n=1, 150, a+=(10 - n)*q; q*=10; write("b064616.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 20 2009

Vec(x*(100*x-9)/((x-1)*(10*x-1)^2) + O(x^100)) \\ Colin Barker, Sep 15 2014

a(n) = a(n-1)+(10-n)*10^(n-1) = (19-n)*A002275(n)-A064617(n) = 10*A002275(n)-A014925(n).

a(n) = 21*a(n-1)-120*a(n-2)+100*a(n-3). G.f.: x*(100*x-9) / ((x-1)*(10*x-1)^2). - Colin Barker, Sep 15 2014

A052245 Expansion of 10x / ((1 - x) (1 - 10*x)^2) in powers of x.

Original entry on oeis.org

0, 10, 210, 3210, 43210, 543210, 6543210, 76543210, 876543210, 9876543210, 109876543210, 1209876543210, 13209876543210, 143209876543210, 1543209876543210, 16543209876543210, 176543209876543210, 1876543209876543210, 19876543209876543210, 209876543209876543210

Offset: 0

Henry Bottomley, Feb 01 2000

This is not the same as A052246. They differ at a(11) and beyond. - Michael Somos, Sep 14 2014

Index entries for linear recurrences with constant coefficients, signature (21,-120,100).

Cf. A014925, A052246.

seq(sum(x*10^x,x=0..a),a=0..100); # Jorge Coveiro, Dec 22 2004
a:=n->sum((10^(n-j)*(n-j)),j=0..n): seq(a(n), n=0..16); # Zerinvary Lajos, Jun 05 2008

concat(0, Vec(-10*x/((x-1)*(10*x-1)^2) + O(x^100))) \\ Colin Barker, Sep 13 2014

a(n) = n*10^n+a(n-1), a(0) = 0; a(n) = ((9n-1)*10^n + 1) * 10 / 81; a(n) = A014925(n)*10.
a(n) = 21*a(n-1)-120*a(n-2)+100*a(n-3). - Colin Barker, Sep 13 2014
G.f.: -10*x / ((x-1)*(10*x-1)^2). - Colin Barker, Sep 13 2014

More terms from Colin Barker, Sep 13 2014

A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

Original entry on oeis.org

1, 11, 811, 3811, 63811, 763811, 3763811, 5103763811, 515103763811, 19515103763811, 6819515103763811, 8146819515103763811, 3808146819515103763811, 7213808146819515103763811, 9807213808146819515103763811, 4939807213808146819515103763811

Offset: 1

Marco Ripà, Apr 04 2024

For any n, a(n) == a(n + 1) (mod 10^len(A038399(n))), where len(k) := number of digits in k. Assuming len(a(n)) > 1, this is a general property of every concatenated sequence with fixed rightmost digits (such as A014925, A061839, A092845, and A104759), as shown in Ripà's book "La strana coda della serie n^n^...^n".

Moreover, assuming n > 1, since A038399(n) is congruent to 11 (mod 20), the convergence speed of A038399(n)^^b (say, V(A038399(n), b) = {2, 1, 1, 1, ...}) is 2 at height 1 and becomes a unit value for any integer b > 1 (see Links). Hence, a(n) is given by A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))), and also by A038399(n)^^len(A038399(n)) (mod 10^len(A038399(n))) since A038399(n)^^len(A038399(n)) == A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))) holds for any n.

			a(8) is given by the rightmost 10 digits of 2113853211^^2113853211 and thus a(8) = 5103763811.
a(9) == a(8) (mod 10^10), i.e., the digits of a(9) end with the digits of a(8) (and then a(9) has 2 more preceding).

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. ISBN 978-88-6178-789-6

Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.
Wikipedia, Tetration.

Cf. A000045 (Fibonacci), A038399, A171882 (tetration), A317824, A317903, A317905.

a(n) = A038399(n)^^(len(A038399(n)) - 1) mod 10^len(A038399(n)), where len(A038399(n)) = ceiling(log_10(A038399(n) + 1)).

cp's OEIS Frontend

A104759 Concatenation of digits of natural numbers from n down to 1.

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A272525 Convolution of nonzero repunits (A002275) with themselves.

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A317824 a(n) = A000422(n)^^A000422(n) (mod 10^len(A000422(n))), where ^^ indicates tetration or hyper-4 (e.g., 3^^4 = 3^(3^(3^3))).

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A317903 a(n) = A038394(n)^^A038394(n) (mod 10^len(A038394(n))), where ^^ indicates tetration or hyper-4 (e.g., 3^^4=3^(3^(3^3))).

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A057138 Add (n mod 10)*10^(n-1) to the previous term, with a(0) = 0.

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A137233 Number of n-digit even numbers.

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A064616 a(n) = (10^n-1)*(91/81)-n*10^n/9.

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A064616 a(n) = (10^n-1)(91/81)-n10^n/9.

A052245 Expansion of 10x / ((1 - x) (1 - 10*x)^2) in powers of x.