A010881 Simple periodic sequence: n mod 12.

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, 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, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The value of the rightmost digit in the base-12 representation of n. - Hieronymus Fischer, Jun 11 2007

Examples

			a(27) = 3 since 27 = 12*2+3.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Crossrefs

Partial sums: A130490. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488, A130489.

Programs

Mathematica

Mod[Range[0, 100], 12] (* Paolo Xausa, Feb 02 2024 *)

PARI

A010881(n)=n%12 \\ M. F. Hasler, Sep 25 2014

Formula

From Hieronymus Fischer, May 31 2007: (Start)

a(n) = n mod 12.

Complex representation: a(n) = (1/12)*(1-r^n)*Sum_{k=1..11} k*Product_{m=1..11, 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_{k=1..11} k*Product_{m=1..11, m<>k} (sin((n-m)*Pi/12))^2.

G.f.: (Sum_{k=1..11} k*x^k)/(1-x^12).

G.f.: x*(11*x^12-12*x^11+1)/((1-x^12)*(1-x)^2). (End)

From Hieronymus Fischer, Jun 11 2007: (Start)

a(n) = (n mod 2)+2*(floor(n/2) mod 6) = A000035(n)+2*A010875(A004526(n)).

a(n) = (n mod 3)+3*(floor(n/3) mod 4) = A010872(n)+3*A010873(A002264(n)).

a(n) = (n mod 4)+4*(floor(n/4) mod 3) = A010873(n)+4*A010872(A002265(n)).

a(n) = (n mod 6)+6*(floor(n/6) mod 2) = A010875(n)+6*A000035(A152467(n)).

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)). (End)

a(A001248(k) + 17) = 6 for k>2. - Reinhard Zumkeller, May 12 2010

a(n) = A034326(n+1)-1. - M. F. Hasler, Sep 25 2014