A130909 Simple periodic sequence (n mod 16).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset

0,3

Comments

The value of the rightmost digit in the base-16 representation of n. Also, the equivalent value of the two rightmost digits in the base-4 representation of n. Also, the equivalent value of the four rightmost digits in the base-2 representation of n.

Links

Table of n, a(n) for n=0..89.

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

Formula

a(n) = n mod 16 = n-16*floor(n/16).

G.f.: g(x) = (Sum_{k=1..15} k*x^k)/(1-x^16).

G.f.: g(x) = x(15x^16-16x^15+1)/((1-x^16)(1-x)^2).

a(n) = A000035(n) + 2*A010877(A004526(n)).

a(n) = A010873(n) + 4*A010873(A002265(n)).

a(n) = A010877(n) + 8*A000035(floor(n/8)).

a(n) = (1/2)*(15 - ( - 1)^n - 2*( - 1)^(b/4) - 4*( - 1)^((b - 2 + 2*( - 1)^(b/4))/8) - 8*( - 1)^((b - 6 + ( - 1)^n + 2*( - 1)^(b/4) + 4*( - 1)^((b - 2 + 2*( - 1)^(b/4))/8))/16)) where b = 2n - 1 + ( - 1)^n.

a(n) = n mod 2+2*(floor(n/2)mod 2)+4*(floor(n/4)mod 2)+8*(floor(n/8)mod 2).

a(n) = (1/2)*(15-(-1)^n-2*(-1)^floor(n/2)-4*(-1)^floor(n/4)-8*(-1)^floor(n/= 8)).

Complex representation: a(n) = (1/16)*(1-r^n)*sum{1<=k<16, k*product{1<=m<16,m<>k, (1-r^(n-m))}} where r=exp(Pi/8*i)=(sqrt(2+sqrt(2))+i*sqrt(2-sqrt(2)))/2 and i=sqrt(-1).

Trigonometric representation: a(n) = 2^22*(sin(n*Pi/16))^2*sum{1<=k<16, k*product{1<=m<16,m<>k, (sin((n-m)*Pi/16))^2}}.

a(n) = (1/2)*(15-(-1)^p(0,n)-2*(-1)^p(1,n)-4*(-1)^p(2,n)-8*(-1)^p(3,n)) where p(k,n) is defined recursively by p(0,n)=n, p(k,n)=1/4*(2*p(k-1,n)-1+(-1)^p(k-1,n)).

Prog

(PARI) a(n)=n%16 \\ Charles R Greathouse IV, Jul 13 2016
(Python)
def A130909(n): return n&15 # Chai Wah Wu, Jan 18 2023

Crossrefs

Cf. partial sums A130910. Other related sequences A010872, A010873, A130481, A130482, A130483, A130486.

See A010877 for a general formula in terms of powers of -1 (for period 2^k).

Sequence in context: A295300 A139179 A262437 * A275993 A160700 A002377

Adjacent sequences: A130906 A130907 A130908 * A130910 A130911 A130912

Keyword

nonn,easy

Author

Hieronymus Fischer, Jun 11 2007

Status

approved