A057139 -id:A057139 - cp's OEIS frontend

A057137 Concatenate next digit at right hand end (where the next digit after 9 is again 0).

0, 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567890, 12345678901, 123456789012, 1234567890123, 12345678901234, 123456789012345, 1234567890123456, 12345678901234567, 123456789012345678, 1234567890123456789, 12345678901234567890, 123456789012345678901

Offset: 0

Henry Bottomley, Aug 12 2000

Also called the triangle of the gods (see Pickover link).

See A037610 for a general formula. - Hieronymus Fischer, Jan 03 2013

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 61.

T. D. Noe and Hieronymus Fischer, Table of n, a(n) for n = 0..200 (terms up to 100 from T. D. Noe)
Clifford Pickover, Triangle of the Gods
Index entries for linear recurrences with constant coefficients, signature (10,0,0,0,0,0,0,0,0,1,-10).

Alternative progression for n >= 10 compared with A007908 and A014824.
Cf. A057138 for reverse. Cf. A010879 (decimal digits).
For primes see A120819.

A057137:=n->floor((137174210/1111111111)*10^n); seq(A057137(n), n=0..20); # Wesley Ivan Hurt, Apr 18 2014

a[n_]:=Floor[137174210/1111111111*10^n]; Array[a,19,0] (* Robert G. Wilson v, Apr 18 2014 *)

A057137(n)=sum(i=1,n,i%10*10^(n-i)) \\ M. F. Hasler, Jan 13 2013

A057137(n)=137174210*10^n\1111111111 \\ M. F. Hasler, Jan 13 2013

def A057137(n): s = '0123456789'; return int((n+1)//10*s + s[:(n+1)%10]) # Ya-Ping Lu, Apr 08 2025

a(n) = 10*(a(n-1)-floor[n/10]) + n = floor[A057139(n)/10^(n-1)].

a(n) = floor((137174210/1111111111)*10^n). - Hieronymus Fischer, Jan 03 2013, corrected by M. F. Hasler, Jan 13 2013

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

A068792 a(n) = (n-1)n^(n-2) + Sum_{i=1..n} (n-i)(n^(n-i-1) + n^(n+i-3)).

Original entry on oeis.org

1, 16, 441, 24336, 2418025, 384473664, 89755965649, 28953439105600, 12345678987654321, 6727499948806851600, 4562491230669011577289, 3769449794266138309731600, 3727710895159027432980276121, 4348096581244536814777202995456, 5907679981266292758213173560296225

Offset: 2

Reinhard Zumkeller, Mar 04 2002

nonn

a(n) is a palindrome in base n representation for all n.

			a(8) = 89755965649 = (1234567654321)OCT;
a(10) = 12345678987654321 = A057139(9);
a(16) = 5907679981266292758213173560296225 = (123456789ABC...987654321)HEX.

G. C. Greubel, Table of n, a(n) for n = 2..215

Cf. A023811, A060072, A062813, A068793.

[((n^(n-1) -1)/(n-1))^2: n in [2..30]]; // G. C. Greubel, Aug 16 2022

Table[((n^(n-1) -1)/(n-1))^2, {n,2,30}] (* G. C. Greubel, Aug 16 2022 *)

def A068792(n): return ((n**(n-1)-1)//(n-1))**2 # Chai Wah Wu, Mar 18 2024

[((n^(n-1) -1)/(n-1))^2 for n in (2..30)] # G. C. Greubel, Aug 16 2022

a(n) = ( (n^(n-1) - 1)/(n-1) )^2.
a(n) = ((A023811(n) - n + 1)/n)*n^(n-1) + A062813(n)/n.
a(n) = A060072(n)^2.

More terms from G. C. Greubel, Aug 16 2022

A261072 Divisors of 1234567890987654321.

Original entry on oeis.org

1, 3, 7, 9, 19, 21, 57, 63, 133, 171, 399, 1197, 928163, 2784489, 6497141, 8353467, 17635097, 19491423, 52905291, 58474269, 123445679, 158715873, 370337037, 1111011111, 1111211111, 3333633333, 7778477777, 10000899999, 21113011109

Offset: 1

Ilya Gutkovskiy, Aug 08 2015

1234567890987654321 = A057139(10) is a palindrome with 48 divisors. See the link with all divisors.
From Wolfdieter Lang, Aug 22 2015: (Start)
The palindromes of this sequence are 1, 3, 7, 9, 171, 1234567890987654321.
1234567890987654321 = 1111011111 * 1111211111 (observed by Jon E. Schoenfield).
The palindromic divisors of 1111011111 are 1, 3, 7, 9 and 171. The only palindromic divisor of 1111211111 is 1. Therefore, of the six palindromes of this sequence only 1234567890987654321 cannot be obtained from the product of the palindromic divisors of 1111011111 with those of 1111211111. (End)

Ilya Gutkovskiy, Table of n, a(n) for n = 1..48
Eric Weisstein's World of Mathematics, Divisor.
Eric Weisstein's World of Mathematics, Palindromic Number.

Cf. A057139, A261245.

[Divisors(1234567890987654321)]; // Vincenzo Librandi, Aug 09 2015

numtheory[divisors](1234567890987654321); # Wesley Ivan Hurt, Aug 11 2015

Mathematica
```
Divisors[1234567890987654321]
```

divisors(1234567890987654321) \\ Wesley Ivan Hurt, Aug 11 2015

cp's OEIS Frontend

A057137 Concatenate next digit at right hand end (where the next digit after 9 is again 0).

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

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A068792 a(n) = (n-1)*n^(n-2) + Sum_{i=1..n} (n-i)*(n^(n-i-1) + n^(n+i-3)).

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A261072 Divisors of 1234567890987654321.

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A068792 a(n) = (n-1)n^(n-2) + Sum_{i=1..n} (n-i)(n^(n-i-1) + n^(n+i-3)).