A010877 a(n) = n mod 8.

0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The rightmost digit in the base-8 representation of n. Also, the equivalent value of the three rightmost digits in the base-2 representation of n. - Hieronymus Fischer, Jun 12 2007

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65536
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Crossrefs

Partial sums: A130486. Other related sequences A130481, A130482, A130483, A130484, A130485.

Cf. A000035, A010887, A010873, A130909, A168181, A244413, A253513.

Programs

Mathematica

Table[Mod[n,8],{n,0,120}]   (* Harvey P. Dale, Apr 21 2011 *)

PARI

vector(100,i,i)%8 \\ Charles R Greathouse IV, Jul 16 2011

Python

def A010877(n): return n&7 # Chai Wah Wu, Jul 09 2022

Formula

Complex representation: a(n) = (1/8)*(1-r^n)*Sum_{k=1..7} k*Product_{m=1..7, m<>k} (1 - r^(n-m)) where r = exp(Pi/4*i) = (1+i)*sqrt(2)/2 and i=sqrt(-1).

Trigonometric representation: a(n) = 256*(sin(n*Pi/8))^2*Sum_{k=1..7} k*Product_{m=1..7, m<>k} (sin((n-m)*Pi/8))^2.

G.f.: g(x) = (Sum_{k=1..7}, k*x^k)/(1-x^8).

Also: g(x) = x(7x^8-8x^7+1)/((1-x^8)(1-x)^2). - Hieronymus Fischer, May 31 2007

a(n) = n mod 2 + 2*(floor(n/2) mod 4) = A000035(n) + 2*A010873(A004526(n)).

a(n) = n mod 4 + 4*(floor(n/4) mod 2) = A010873(n) + 4*A000035(A002265(n)).

a(n) = n mod 2 + 2*(floor(n/2) mod 2) + 4*(floor(n/4) mod 2) = A000035(n) + 2*A000035(A004526(n)) + 4*A000035(A002265(n)). - Hieronymus Fischer, Jun 12 2007

a(n) = (1/2)*(7 - (-1)^n - 2*(-1)^(b/4) - 4*(-1)^((b - 2 + 2*(-1)^(b/4))/8)) where b = 2n - 1 + (-1)^n. - Hieronymus Fischer, Jun 12 2007

General formula for period 2^k: a(n) = (1/2)*(2^k - 1 - Sum_{j=0..k-1} 2^j*(-1)^p(j,n)) where p(j,n) is defined recursively by p(0,n)=n, p(j,n) = (1/4)*(2*p(j-1,n) - 1 + (-1)^p(j-1,n)). - Hieronymus Fischer, Jun 14 2007

a(n) = floor(1234567/99999999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013

a(n) = floor(48913/2396745*8^(n+1)) mod 8. - Hieronymus Fischer, Jan 04 2013

Extensions

Formula section re-edited for better readability by Hieronymus Fischer