A034948 -id:A034948 - cp's OEIS frontend

A010879 Final digit of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0

Offset: 0

N. J. A. Sloane

Also decimal expansion of 137174210/1111111111 = 0.1234567890123456789012345678901234... - Jason Earls, Mar 19 2001
In general the base k expansion of A062808(k)/A048861(k) (k>=2) will produce the numbers 0,1,2,...,k-1 repeated with period k, equivalent to the sequence n mod k. The k-digit number in base k 123...(k-1)0 (base k) expressed in decimal is A062808(k), whereas A048861(k) = k^k-1. In particular, A062808(10)/A048861(10)=1234567890/9999999999=137174210/1111111111.
a(n) = n^5 mod 10. - Zerinvary Lajos, Nov 04 2009

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).
Index entries for sequences related to final digits of numbers

Cf. A034948, A059988, A048861, A062808, A086457, A086458.
Cf. A008959, A008960, A070514. - Doug Bell, Jun 15 2015
Partial sums: A130488. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486, A130487.

a010879 = (`mod` 10)
a010879_list = cycle [0..9]  -- Reinhard Zumkeller, Mar 26 2012

[n mod(10): n in [0..90]]; // Vincenzo Librandi, Jun 17 2015

A010879 := proc(n)
    n mod 10 ;
end proc: # R. J. Mathar, Jul 12 2013

Table[10*FractionalPart[n/10], {n, 1, 300}] (* Enrique Pérez Herrero, Jul 30 2009 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 1, 2, 3, 4, 5, 6, 7, 8, 9},81] (* Ray Chandler, Aug 26 2015 *)
PadRight[{},100,Range[0,9]] (* Harvey P. Dale, Oct 04 2021 *)

a(n)=n%10 \\ Charles R Greathouse IV, Jun 16 2011

def a(n): return n % 10 # Martin Gergov, Oct 17 2022

[power_mod(n,5,10)for n in range(0, 81)] # Zerinvary Lajos, Nov 04 2009

a(n) = n mod 10.
Periodic with period 10.
From Hieronymus Fischer, May 31 and Jun 11 2007: (Start)
Complex representation: a(n) = 1/10*(1-r^n)*sum{1<=k<10, k*product{1<=m<10,m<>k, (1-r^(n-m))}} where r=exp(Pi/5*i) and i=sqrt(-1).
Trigonometric representation: a(n) = (256/5)^2*(sin(n*Pi/10))^2 * sum{1<=k<10, k*product{1<=m<10,m<>k, (sin((n-m)*Pi/10))^2}}.
G.f.: g(x) = (Sum_{k=1..9} k*x^k)/(1-x^10) = -x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8) / ( (x-1) *(1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) ).
Also: g(x) = x*(9*x^10-10*x^9+1)/((1-x^10)*(1-x)^2).
a(n) = n mod 2+2*(floor(n/2)mod 5) = A000035(n) + 2*A010874(A004526(n)).
Also: a(n) = n mod 5+5*(floor(n/5)mod 2) = A010874(n)+5*A000035(A002266(n)). (End)
a(n) = 10*{n/10}, where {x} means fractional part of x. - Enrique Pérez Herrero, Jul 30 2009
a(n) = n - 10*A059995(n). - Reinhard Zumkeller, Jul 26 2011
a(n) = n^k mod 10, for k > 0, where k mod 4 = 1. - Doug Bell, Jun 15 2015

Formula section edited for better readability by Hieronymus Fischer, Jun 13 2012

A036663 Decimal expansion of 1/98019801.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 0, 4, 0, 5, 0, 6, 0, 6, 0, 6, 0, 7, 0, 8, 0, 8, 0, 8, 0, 9, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 1, 4, 1, 4, 1, 5, 1, 6, 1, 6, 1, 6, 1, 7, 1, 8, 1, 8, 1, 8, 1, 9, 2, 0, 2, 0, 2, 0, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 4, 2

Offset: 0

Marvin Ray Burns

Cf. A034948, A036664, A036665.

Join[{0,0,0,0,0,0,0},RealDigits[1/98019801 ,10,120][[1]]] (* Harvey P. Dale, Jul 01 2019 *)

A036664 Decimal expansion of 1/980198019801.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 5, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 7, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 9, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 6, 1

Offset: 0

Marvin Ray Burns

Cf. A034948, A036663, A036665.

Join[PadRight[{},11,0],RealDigits[1/980198019801,10,120][[1]]] (* Harvey P. Dale, Jul 28 2023 *)

A036665 Decimal expansion of 1/9801980198019801.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 5, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 7, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 9, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1

Offset: 0

Marvin Ray Burns

Cf. A034948, A036663, A036664.

A227093 Decimal expansion of 1/9899.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, 1, 3, 2, 1, 3, 4, 5, 5, 9, 0, 4, 6, 3, 6, 8, 3, 2, 0, 0, 3, 2, 3, 2, 6, 4, 9, 7, 6, 2, 6, 0, 2, 2, 8, 3, 0, 5, 8, 8, 9, 4, 8, 3, 7, 8, 6, 2, 4, 1, 0, 3, 4, 4, 4, 7, 9, 2, 4, 0, 3, 2, 7, 3, 0, 5, 7, 8, 8, 4, 6, 3, 4, 8, 1, 1, 5, 9, 7, 1, 3, 1, 0, 2, 3, 3, 3, 5, 6, 9, 0, 4, 7

Offset: 0

Roland Kneer, Jul 01 2013

Group the terms 2 by 2 to get the first 11 Fibonacci numbers (A000045): 00 01 01 02 03 05 08 13 21 34 55 (89, 144, 233, ...).

			0.00010102030508132134559046368320032326497626022830588...

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A000045, A021093, A034948.

Maple
```
Digits := 140; evalf(1/9899);
```

First[RealDigits[1/9899, 10, 100, -1]] (* Paolo Xausa, Jun 16 2024 *)

Equals Sum_{i>=0} Fibonacci(i)/100^(i+1).

A384627 Decimal expansion of 1/998001.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 5, 0, 0, 6, 0, 0, 7, 0, 0, 8, 0, 0, 9, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0, 1, 6, 0, 1, 7, 0, 1, 8, 0, 1, 9, 0, 2, 0, 0, 2, 1, 0, 2, 2, 0, 2, 3, 0, 2, 4, 0, 2, 5, 0, 2, 6, 0, 2, 7, 0, 2, 8, 0, 2, 9

Offset: 0

Paolo Xausa, Jun 05 2025

The decimals of this constant contain every 3-digit number, from 000 to 999, in order, except for 998.

Periodic with period 2997.

			0.0000010020030040050060070080090100110120130140150160170...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Brady Haran and James Grime, 998,001 and its Mysterious Recurring Decimals, YouTube Numberphile video, 2012.

Cf. A021085, A034948.

Mathematica
```
First[RealDigits[1/999^2, 10, 100, -1]]
```

Equals 1/(999^2).

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A010879 Final digit of n.

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A036663 Decimal expansion of 1/98019801.

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A036664 Decimal expansion of 1/980198019801.

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A036665 Decimal expansion of 1/9801980198019801.

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A227093 Decimal expansion of 1/9899.

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A384627 Decimal expansion of 1/998001.

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