A010881 - OEIS (original) (raw)

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

COMMENTS

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

LINKS

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

FORMULA

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)

EXAMPLE

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

MATHEMATICA

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