A010887 -id:A010887 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

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

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

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

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

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

Formula section re-edited for better readability by Hieronymus Fischer

A168183 Numbers that are not multiples of 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Reinhard Zumkeller, Nov 30 2009

It seems that, for any n >= 1, there exists no positive integer z such that digit_sum(z) = digit_sum(a(n)+z). - Max Lacoma, Sep 19 2019. Giovanni Resta: this follows immediately from the well-known fact that sod(x) == x (mod 9).

Ivan Panchenko, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Complement of A008591.

Cf. A010887, A010888, A109012, A168182, A248375.

a168183 n = a168183_list !! (n-1)
a168183_list = [1..8] ++ map (+ 9) a168183_list
-- Reinhard Zumkeller, Mar 04 2014

[n+Floor((n-1)/8) : n in [1..100]]; // Wesley Ivan Hurt, Sep 12 2015

A168183:=n->n+floor((n-1)/8): seq(A168183(n), n=1..100); # Wesley Ivan Hurt, Sep 12 2015

Select[Table[n,{n,200}],Mod[#,9]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
With[{nn=81},Complement[Range[nn],9Range[Floor[nn/9]]]] (* Harvey P. Dale, Sep 07 2011 *)

is(n)=!!(n%9) \\ Charles R Greathouse IV, Sep 02 2015

a(n)=(9*n-1)\8 \\ Charles R Greathouse IV, Sep 02 2015

from gmpy2 import f_mod
[n for n in range(100) if f_mod(n,9)] # Bruno Berselli, Dec 05 2016

A168182(a(n)) = 1.
A010888(a(n)) = A010887(n-1).
A109012(a(n)) < 9.
From Wesley Ivan Hurt, Sep 12 2015: (Start)
a(n) = a(n-1) + a(n-8) - a(n-9), n>9.
a(n) = n + floor((n-1)/8). (End)
From Philippe Deléham, Dec 05 2016: (Start)
a(n) = 1 + A248375(n-1).
G.f.: x*(1-x^9)/((1-x)^2*(1-x^8)). (End)
E.g.f.: 1 + (1/8)*(-cos(x) + (-5+9*x)*cosh(x) - 2*cos(x/sqrt(2))*cosh(x/sqrt(2)) + sin(x) + (-4+9*x)*sinh(x) + 2*sin(x/sqrt(2))*(sqrt(2)*cosh(x/sqrt(2)) + sinh(x/sqrt(2)))). - Stefano Spezia, Sep 20 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3) + 3*cosec(2*Pi/9) - 3*tan(Pi/18)) * Pi/27. - Amiram Eldar, May 11 2025

A177034 Decimal expansion of (9280+3*sqrt(13493990))/14165.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 01 2010

Continued fraction expansion of (9280+3*sqrt(13493990))/14165 is A010887.

			(9280+3*sqrt(13493990))/14165 = 1.43312742655589912842...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A177035 (decimal expansion of sqrt(13493990)), A010887 (repeat 1, 2, 3, 4, 5, 6, 7, 8).

First@RealDigits@N[(9280 + 3 Sqrt[13493990])/14165, 200] (* Vincenzo Librandi, Dec 29 2016 *)

A181753 Universal sequence of period 56 which contains every 3-subset of {1,2,...,8} exactly once.

Original entry on oeis.org

1, 3, 5, 6, 7, 2, 5, 6, 8, 2, 3, 4, 7, 2, 3, 5, 7, 8, 1, 4, 7, 8, 2, 4, 5, 6, 1, 4, 5, 7, 1, 2, 3, 6, 1, 2, 4, 6, 7, 8, 3, 6, 7, 1, 3, 4, 5, 8, 3, 4, 6, 8, 1, 2, 5, 8, 1, 3, 5, 6, 7, 2, 5, 6, 8, 2, 3, 4, 7, 2, 3, 5, 7, 8, 1, 4, 7, 8, 2, 4, 5, 6, 1, 4, 5, 7, 1, 2, 3, 6, 1, 2, 4, 6, 7, 8, 3, 6, 7, 1, 3, 4, 5, 8, 3, 4, 6, 8, 1, 2, 5, 8, 1, 3, 5, 6, 7, 2, 5, 6, 8, 2, 3, 4, 7, 2, 3, 5, 7, 8, 1, 4, 7, 8, 2, 4, 5, 6, 1, 4, 5, 7, 1, 2, 3, 6, 1, 2, 4, 6, 7, 8, 3, 6, 7, 1, 3, 4, 5, 8, 3, 4, 6, 8, 1, 2, 5, 8

Offset: 1

Susanna Cuyler, Nov 14 2010

Each successive block of length 7 is obtained by adding 5 mod 8 to the previous block.

			The period is 1356725 6823472 3578147 8245614 5712361 2467836 7134583 4681258.

G. Hurlbert, On universal cycles for k-subsets of an n-element set, SIAM J. Discrete Math., 7 (1994), 598-604.
A. Leitner and A. Godbole, Universal cycles of classes of restricted words, Discrete Math., 310 (2010) 3303-3309.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1).

Cf. A010887.

a181753 n = a181753_list !! (n-1)
a181753_list = concat $ iterate
               (map ((+ 1) . flip mod 8 . (+ 4))) [1,3,5,6,7,2,5]
-- Reinhard Zumkeller, Nov 09 2014

From Chai Wah Wu, Jun 13 2020: (Start)
a(n) = a(n-1) - a(n-7) + a(n-8) - a(n-14) + a(n-15) - a(n-21) + a(n-22) - a(n-28) + a(n-29) - a(n-35) + a(n-36) - a(n-42) + a(n-43) - a(n-49) + a(n-50) for n > 50.
G.f.: x*(-8*x^49 + 3*x^48 + 3*x^47 + x^46 - 7*x^45 + 2*x^44 + 2*x^43 - 7*x^42 - 2*x^41 + 6*x^40 + 2*x^39 - 6*x^38 + 4*x^37 - 4*x^36 - 6*x^35 + x^34 + x^33 + 3*x^32 - 5*x^31 + 6*x^30 - 2*x^29 - 5*x^28 - 4*x^27 + 4*x^26 + 4*x^25 - 4*x^24 - 4*x^21 - x^20 - x^19 + 5*x^18 - 3*x^17 + 2*x^16 - 6*x^15 - 3*x^14 + 2*x^13 + 2*x^12 - 2*x^11 - 2*x^10 + 4*x^9 - 4*x^8 - 2*x^7 - 3*x^6 + 5*x^5 - x^4 - x^3 - 2*x^2 - 2*x - 1)/(x^50 - x^49 + x^43 - x^42 + x^36 - x^35 + x^29 - x^28 + x^22 - x^21 + x^15 - x^14 + x^8 - x^7 + x - 1). (End)

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A010877 a(n) = n mod 8.

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A168183 Numbers that are not multiples of 9.

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A177034 Decimal expansion of (9280+3*sqrt(13493990))/14165.

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A181753 Universal sequence of period 56 which contains every 3-subset of {1,2,...,8} exactly once.

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