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

0,3

Comments

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

Links

Table of n, a(n) for n=0..83.

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

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A130024 A131232 A297238 * A382940 A190600 A053832

Adjacent sequences: A010878 A010879 A010880 * A010882 A010883 A010884

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved