A138531 -id:A138531 - cp's OEIS frontend

A180598 Digital root of 8n.

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

Offset: 0

Odimar Fabeny, Sep 10 2010

Period of 9. - Robert G. Wilson v, Sep 20 2010

Essentially the same as A138531. - R. J. Mathar, Jul 09 2011

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

Cf. A007953, A010888, A180592 - A180599.

f[n_] := Mod[8 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)

G.f.: -x*(9*x^8+x^7+2*x^6+3*x^5+4*x^4+5*x^3+6*x^2+7*x+8)/((x-1)*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Aug 19 2012

a(n) = A010888(8*n). - R. J. Mathar, Aug 28 2025

More terms from Robert G. Wilson v, Sep 20 2010

A309818 Digits of the 10-adic integer (987654321/(1-10^9))^(1/3).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 18 2019

x = ...113171709287171280192602158871517841.

x^3 = ...987654321987654321987654321987654321.

			          1^3 == 1         (mod 10).
         41^3 == 21        (mod 10^2).
        841^3 == 321       (mod 10^3).
       7841^3 == 4321      (mod 10^4).
      17841^3 == 54321     (mod 10^5).
     517841^3 == 654321    (mod 10^6).
    1517841^3 == 7654321   (mod 10^7).
   71517841^3 == 87654321  (mod 10^8).
  871517841^3 == 987654321 (mod 10^9).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Digits of the 10-adic integer (987654321/(1-10^9))^(1/k): this sequence (k=3), A309819 (k=7), A309820 (k=9).

Cf. A010888, A138531, A309821, A309824.

N=100; M=987654321/(1-10^9); Vecrev(digits(lift(chinese(Mod((M+O(2^N))^(1/3), 2^N), Mod((M+O(5^N))^(1/3), 5^N)))), N)

A309819 Digits of the 10-adic integer (987654321/(1-10^9))^(1/7).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 18 2019

x = ...612123123752627210431674758928842961.

x^7 = ...987654321987654321987654321987654321.

			          1^7 == 1         (mod 10).
         61^7 == 21        (mod 10^2).
        961^7 == 321       (mod 10^3).
       2961^7 == 4321      (mod 10^4).
      42961^7 == 54321     (mod 10^5).
     842961^7 == 654321    (mod 10^6).
    8842961^7 == 7654321   (mod 10^7).
   28842961^7 == 87654321  (mod 10^8).
  928842961^7 == 987654321 (mod 10^9).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Digits of the 10-adic integer (987654321/(1-10^9))^(1/k): A309818 (k=3), this sequence (k=7), A309820 (k=9).

Cf. A010888, A138531, A309822, A309825.

N=100; M=987654321/(1-10^9); Vecrev(digits(lift(chinese(Mod((M+O(2^N))^(1/7), 2^N), Mod((M+O(5^N))^(1/7), 5^N)))), N)

A309820 Digits of the 10-adic integer (987654321/(1-10^9))^(1/9).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 18 2019

x = ...338737664587063665050177076914840881.

x^9 = ...987654321987654321987654321987654321.

			          1^9 == 1         (mod 10).
         81^9 == 21        (mod 10^2).
        881^9 == 321       (mod 10^3).
        881^9 == 4321      (mod 10^4).
      40881^9 == 54321     (mod 10^5).
     840881^9 == 654321    (mod 10^6).
    4840881^9 == 7654321   (mod 10^7).
   14840881^9 == 87654321  (mod 10^8).
  914840881^9 == 987654321 (mod 10^9).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Digits of the 10-adic integer (987654321/(1-10^9))^(1/k): A309818 (k=3), A309819 (k=7), this sequence (k=9).

Cf. A010888, A138531, A309823, A309826.

N=100; M=987654321/(1-10^9); Vecrev(digits(lift(chinese(Mod((M+O(2^N))^(1/9), 2^N), Mod((M+O(5^N))^(1/9), 5^N)))), N)

A262734 Period 16: repeat (1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 29 2015

Decimal expansion of 111111112/900000009.

For n which lies in the interval [16*(k-1), 8*(2*k-1)], where k>0 -> pattern {1, 2, 3, 4, 5, 6, 7, 8, 9}; for n which lies in the interval [16*k - 7, 16*k - 1], where k>0 -> pattern {8, 7, 6, 5, 4, 3, 2}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,-1,1).

Cf. A158289, A177274, A010889, A138531, A181975, A199264, A068073, A028356.

&cat[[1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2]: n in [0..10]]; // Vincenzo Librandi, Sep 29 2015

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, -1, 1}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, 120] (* Vincenzo Librandi, Sep 29 2015 *)

Vec(-(2*x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)*(x^8+1)) + O(x^100)) \\ Colin Barker, Sep 29 2015

111111112/900000009. \\ Altug Alkan, Sep 29 2015

vector(200, n, default(realprecision, n+2); floor(111111112/900000009*10^n)%10) \\ Altug Alkan, Nov 12 2015

-1 + a(16*(k - 1)) = -2 + a(8*k + 3*(-1)^k - 4) = -3 + a(2*(4*k + (-1)^k - 2)) = -4 + a(8*k + (-1)^k - 4) = -5 + a(4*(2*k - 1)) = -6 + a(8*k - (-1)^k - 4) = -7 + a(-2*(-4*k + (-1)^k + 2)) = -8 + a(8*k - 3*(-1)^k - 4) = -9 + a(8*(2*k - 11)) = 0, for k>0.
a(0) = 1, a(n) = a(n+1) - 1, for 16*(k - 1) <= n < 8*(2*k - 1), and a(n) = a(n + 1) + 1, for 8*(2*k - 1) <= n < 16*k, where k>0.
From Colin Barker, Sep 29 2015: (Start)
a(n) = a(n-1) - a(n-8) + a(n-9) for n>8.
G.f.: -(2*x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1) / ((x-1)*(x^8+1)). (End)

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A180598 Digital root of 8n.

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A309818 Digits of the 10-adic integer (987654321/(1-10^9))^(1/3).

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A309819 Digits of the 10-adic integer (987654321/(1-10^9))^(1/7).

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A309820 Digits of the 10-adic integer (987654321/(1-10^9))^(1/9).

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A262734 Period 16: repeat (1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2).

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