A180592 -id:A180592 - cp's OEIS frontend

A180599 Zero followed by infinitely many 9's.

0, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Odimar Fabeny, Sep 10 2010

Another interpretation: A real number with an infinitesimally small difference from the integer 1 which is used to test the precision of calculating devices. - John W. Nicholson, Feb 01 2012

a(n) is also the number of n-digit positive repdigit numbers (A010785). - Stefano Spezia, Aug 15 2020

			Viewed as a real number: For a TI-89, entering 1.-10^-12 yields .999999999999; however, 1.-10^-13 yields 1. - _John W. Nicholson_, Feb 01 2012

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

Cf. A007953, A010785, A010888, A057427, A180592, A180594, A180595, A180596, A180597, A180598.

a[n_] := Mod[9 n - 1, 9] + 1; a[0] = 0; Array[a, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)
PadRight[{0},120,9] (* Harvey P. Dale, May 03 2019 *)

a(n)=if(n,9,0) \\ Charles R Greathouse IV, Jan 04 2013

a(0) = 0, a(n) = 9 for n > 0.
a(n) = 9 * A057427(n).
a(n) = A010888(9*n), where A010888 is the digital root.
From Robert Israel, Dec 16 2014: (Start)
G.f.: 9*x/(1 - x).
E.g.f.: 9*(exp(x) - 1). (End)

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

Definition changed by N. J. A. Sloane, Feb 04 2012

A180597 Digital root of 7n.

Original entry on oeis.org

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

Offset: 0

Odimar Fabeny, Sep 10 2010

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

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

Cf. A010888, A008589.
Cf. A180592, A180594, A180595, A180596, A180592, A180593, A180594, A180595, A180596, A180598, A180599.
Column k = 7 in A353109.

f[n_] := Mod[7 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)

a(n)=7*n%9 \\ Charles R Greathouse IV, Aug 22 2014

G.f.: x*(7 + 5*x + 3*x^2 + x^3 + 8*x^4 + 6*x^5 + 4*x^6 + 2*x^7 + 9*x^8)/(1 - x^9). - Stefano Spezia, Apr 21 2022

a(n) = A010888(A008589(n)). - Michel Marcus, Apr 21 2022

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

A180598 Digital root of 8n.

Original entry on oeis.org

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

A180593 Digital root of 3n.

Original entry on oeis.org

0, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6

Offset: 0

Odimar Fabeny, Sep 10 2010

Decimal expansion of 41/1110. - Enrique Pérez Herrero, Nov 13 2021

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

Cf. A008585 (3*n), A010888 (digital root), A002277.

Cf. A180592, A180594, A180595, A180596, A180597, A180598, A180599.

digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&,n]; Table[If[n==0,0,digitalRoot[3*n]], {n,0,200}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)
LinearRecurrence[{0,0,1},{0,3,6,9},120] (* Harvey P. Dale, Sep 03 2020 *)

a(n+1) = 3*A010882(n). - Reinhard Zumkeller, Oct 25 2010
G.f.: (-3*(1 + 2*x + 3*x^2))/(-1 + x^3) for n>0. - Alexander R. Povolotsky, Jun 13 2012
a(n) = A010888(A002277(n)). - Enrique Pérez Herrero, Nov 24 2022
a(n) = A010888(A008585(n)). - Michel Marcus, Nov 24 2022

Edited by N. J. A. Sloane, Sep 23 2010

A180594 Digital root of 4n.

Original entry on oeis.org

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

Offset: 0

Odimar Fabeny, Sep 10 2010

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

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

Cf. A007953, A010888, A180592.

Cf. A180592, A180593, A180595, A180596, A180597, A180598, A180599.

f[n_] := Mod[4 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)
Join[{0}, ReplaceAll[Table[Mod[4n, 9], {n, 99}], {0 -> 9}]] (* Alonso del Arte, Sep 23 2012 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,1},{0,4,8,3,7,2,6,1,5,9},120] (* or *) PadRight[ {0},120,{9,4,8,3,7,2,6,1,5}] (* Harvey P. Dale, Aug 09 2022 *)

G.f.: -x*(9*x^8+5*x^7+x^6+6*x^5+2*x^4+7*x^3+3*x^2+8*x+4)/((x-1)*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Sep 23 2012

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

A180595 Digital root of 5n.

Original entry on oeis.org

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

Offset: 0

Odimar Fabeny, Sep 10 2010

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

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

Cf. A007953, A010888, A180592, A180594.

Cf. A180592, A180593, A180594, A180596, A180597, A180598, A180599.

f[n_] := Mod[5 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)
Join[{0}, ReplaceAll[Table[Mod[5n, 9], {n, 99}], {0 -> 9}]] (* Alonso del Arte, Sep 23 2012 *)
PadRight[{0},120,{9,5,1,6,2,7,3,8,4}] (* Harvey P. Dale, Jan 10 2024 *)

G.f.: -x*(9*x^8+4*x^7+8*x^6+3*x^5+7*x^4+2*x^3+6*x^2+x+5)/((x-1)*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Sep 23 2012

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

A180596 Digital root of 6n.

Original entry on oeis.org

0, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3

Offset: 0

Odimar Fabeny, Sep 10 2010

Period of 3. - Robert G. Wilson v, Sep 20 2010

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

Cf. A010888, A008588.
Cf. A180592, A180593, A180594, A180595, A180597, A180598, A180599.
Column k = 6 in A353109.

f[n_] := Mod[6 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)
PadRight[{0},120,{9,6,3}] (* Harvey P. Dale, Dec 18 2012 *)

G.f.: 3*x*(2 + x + 3*x^2)/(1 - x^3). - Stefano Spezia, Apr 21 2022

a(n) = A010888(A008588(n)). - Michel Marcus, Apr 24 2022

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

A353109 Array read by antidiagonals: A(n, k) is the digital root of n*k with n >= 0 and k >= 0.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Apr 24 2022

			The array begins:
    0, 0, 0, 0, 0, 0, 0, 0, ...
    0, 1, 2, 3, 4, 5, 6, 7, ...
    0, 2, 4, 6, 8, 1, 3, 5, ...
    0, 3, 6, 9, 3, 6, 9, 3, ...
    0, 4, 8, 3, 7, 2, 6, 1, ...
    0, 5, 1, 6, 2, 7, 3, 8, ...
    0, 6, 3, 9, 6, 3, 9, 6, ...
    0, 7, 5, 3, 1, 8, 6, 4, ...
    ...

Cf. A003991, A004247, A010888, A056992 (diagonal), A073636, A139413, A180592, A180593, A180594, A180595, A180596, A180597, A180598, A180599, A303296, A336225, A353128 (antidiagonal sums), A353933, A353974 (partial sum of the main diagonal).

A[i_,j_]:=If[i*j==0,0,1+Mod[i*j-1,9]];Flatten[Table[A[n-k,k],{n,0,12},{k,0,n}]]

T(n,k) = if (n && k, (n*k-1)%9+1, 0); \\ Michel Marcus, May 12 2022

A(n, k) = A010888(A004247(n, k)).

A(n, k) = A010888(A003991(n, k)) for n*k > 0.

A139413 Triangle read by rows: row n gives the numbers A010888(n*k) for k = 1..n.

Original entry on oeis.org

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

Offset: 1

Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 19 2008

			Triangle begins:
1
2 4
3 6 9
4 8 3 7
5 1 6 2 7
6 3 9 6 3 9
7 5 3 1 8 6 4
8 7 6 5 4 3 2 1
9 9 9 9 9 9 9 9 9
...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A007953 (column 1), A180592 - A180599 (columns 2 - 9), A010888, A038194.

A010888:=proc(n) return ((n-1) mod 9)+1; end:
for n from 1 to 9 do for k from 1 to n do printf("%d ",A010888(n*k)); od: printf("\n"): od: # Nathaniel Johnston, Apr 21 2011

Table[FixedPoint[Total@ IntegerDigits[#] &, n k], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Oct 31 2021 *)

More terms and several terms corrected by Nathaniel Johnston, Apr 21 2011

A269226 Period 6: repeat [3, 9, 6, 6, 9, 3].

Original entry on oeis.org

3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3

Offset: 1

Peter M. Chema, Jul 11 2016

The palindromic sequence arising when the digital root of n alternates diagonally in opposite directions on a square grid.  This is the sequence of 3-6-9 appearing every third column on a square grid when A010888 (digital root of n) alternates in both directions diagonally. Other columns are the digital root of 2^n: {1, 2, 4, 8, 7, 5}, or in its opposite direction 5^n: {5,7,8,4,2,1}.  All diagonals parallel to the digital roots of n are also {1,2,3,4,5,6,7,8,9} or {9,8,7,6,5,4,3,2,1}.
See the link below for a visual illustration.
This sequence also arises when A180592 (digital root of 2n) is substituted for A010888.
Decimal expansion of 40070/10101. - David A. Corneth, Jul 12 2016

Peter M. Chema, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1).

Cf. A010888, A180592.

a(n)=[3, 3, 9, 6, 6, 9][n%6+1] \\ Charles R Greathouse IV, Jul 12 2016

a(n+1) = digital root of 5^n - 2^n.
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) = a(n-6). - Charles R Greathouse IV, Jul 12 2016
a(n) = (12 - 3*cos(n*Pi/3) - 3*cos(2*n*Pi/3) - sqrt(3)*sin(n*Pi/3) - 3*sqrt(3)*sin(2*n*Pi/3))/2. - Wesley Ivan Hurt, Oct 05 2018

cp's OEIS Frontend

A180599 Zero followed by infinitely many 9's.

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A180597 Digital root of 7n.

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A180598 Digital root of 8n.

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A180593 Digital root of 3n.

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A180594 Digital root of 4n.

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A180595 Digital root of 5n.

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A180596 Digital root of 6n.

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