%I #22 Feb 02 2024 06:54:36
%S 0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,
%T 8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,
%U 3,4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11
%N Simple periodic sequence: n mod 12.
%C The value of the rightmost digit in the base-12 representation of n. - _Hieronymus Fischer_, Jun 11 2007
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,1).
%F a(n) = n mod 12. Complex representation: a(n)=(1/12)*(1-r^n)*sum{1<=k<12, k*product{1<=m<12,m<>k, (1-r^(n-m))}} where r=exp(Pi/6*i)=(sqrt(3)+i)/2 and i=sqrt(-1). Trigonometric representation: a(n)=(512/3)^2*(sin(n*Pi/12))^2*sum{1<=k<12, k*product{1<=m<12,m<>k, (sin((n-m)*Pi/12))^2}}. G.f.: g(x)=(sum{1<=k<12, k*x^k})/(1-x^12). Also: g(x)=x(11x^12-12x^11+1)/((1-x^12)(1-x)^2). - _Hieronymus Fischer_, May 31 2007
%F a(n) = n mod 2+2*(floor(n/2)mod 6)=A000035(n)+2*A010875(A004526(n)). Also: a(n)=n mod 3+3*(floor(n/3)mod 4)=A010872(n)+3*A010873(A002264(n)). Also: a(n)=n mod 4+4*(floor(n/4)mod 3)=A010873(n)+4*A010872(A002265(n)). Also: a(n)=n mod 6+6*(floor(n/6)mod 2)=A010875(n)+6*A000035(floor(n/6)). Also: a(n)=n mod 2+2*(floor(n/2)mod 2+4*(floor(n/4)mod 3)=A000035(n)+2*A000035(A004526(n))+4*A010872(A002265(n)). - _Hieronymus Fischer_, Jun 11 2007
%F a(A001248(k) + 17) = 6 for k>2. - _Reinhard Zumkeller_, May 12 2010
%F a(n) = A034326(n+1)-1. - _M. F. Hasler_, Sep 25 2014
%e a(27)=3 since 27=12*2+3.
%t Mod[Range[0, 100], 12] (* _Paolo Xausa_, Feb 02 2024 *)
%o (PARI) A010881(n)=n%12 \\ _M. F. Hasler_, Sep 25 2014
%Y Partial sums: A130490. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488, A130489.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_.