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A010879
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Final digit of n.
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128
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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Table of n, a(n) for n=0..80.
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
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FORMULA
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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{1<=k<10, 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(9x^10-10x^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
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MAPLE
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A010879 := proc(n)
n mod 10 ;
end proc: # R. J. Mathar, Jul 12 2013
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MATHEMATICA
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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 *)
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PROG
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(Sage) [power_mod(n, 5, 10)for n in range(0, 81)] # Zerinvary Lajos, Nov 04 2009
(PARI) a(n)=n%10 \\ Charles R Greathouse IV, Jun 16 2011
(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
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CROSSREFS
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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.
Sequence in context: A285094 A134778 A118943 * A179636 A217657 A175419
Adjacent sequences: A010876 A010877 A010878 * A010880 A010881 A010882
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KEYWORD
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nonn,base,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Formula section edited for better readability by Hieronymus Fischer, Jun 13 2012
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STATUS
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approved
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