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

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy

Comments

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

Links

• 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

Crossrefs

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.

Programs

Haskell

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

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Sage

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

Formula

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

Extensions

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