A180599 -id:A180599 - cp's OEIS frontend

A180592 Digital root of 2n.

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

Offset: 0

Odimar Fabeny, Sep 10 2010

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

Also digital root of A002276(n). - Enrique Pérez Herrero, Nov 05 2022

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

Cf. A007953, A010888, A180593, A180594, A180595, A180596, A180597, A180598, A180599.

f[n_] := Mod[2 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)

a(n)=if(n,(2*n-1)%9+1,0) \\ Charles R Greathouse IV, Aug 25 2014

From R. J. Mathar, Nov 02 2010: (Start)
a(n) = A010888(2*n).
a(n) = a(n-9), n > 9.
G.f.: -x*(2 + 4*x + 6*x^2 + 8*x^3 + x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 9*x^8) / ( (x-1)*(1 + x + x^2)*(x^6 + x^3 + 1) ). (End)

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

Keyword:base and formulas from R. J. Mathar, Nov 02 2010

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

A337127 Table with 10 columns read by rows: T(n, k) is the number of n-digit positive integers with exactly k distinct base 10 digits (0 < k <= 10).

Original entry on oeis.org

9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 81, 0, 0, 0, 0, 0, 0, 0, 0, 9, 243, 648, 0, 0, 0, 0, 0, 0, 0, 9, 567, 3888, 4536, 0, 0, 0, 0, 0, 0, 9, 1215, 16200, 45360, 27216, 0, 0, 0, 0, 0, 9, 2511, 58320, 294840, 408240, 136080, 0, 0, 0, 0, 9, 5103, 195048, 1587600, 3810240, 2857680, 544320, 0, 0, 0

Offset: 1

Stefano Spezia, Aug 17 2020

			The table T(n, k) begins:
9     0      0       0       0       0  0  0  0  0
9    81      0       0       0       0  0  0  0  0
9   243    648       0       0       0  0  0  0  0
9   567   3888    4536       0       0  0  0  0  0
9  1215  16200   45360   27216       0  0  0  0  0
9  2511  58320  294840  408240  136080  0  0  0  0
...

Index entries for sequences related to digits.

Cf. A010785, A031955, A031962, A031969, A031987, A116670, A171102, A218019, A219743, A220076.

Cf. A010734, A048993, A052268 (row sums), A073531 (diagonal), A180599 (k = 1), A335843 (k = 2), A337313 (k = 3).

T[n_,k_]:=9Pochhammer[11-k,k-1]/k!*n!*Coefficient[Series[(Exp[x]-1)^k,{x,0,n}],x,n]; Table[T[n,k],{n,7},{k,10}]//Flatten

T(n, k) = 9*Pochhammer(11-k, k-1)*n! * [x^n] (exp(x) - 1)^k/k!.
T(n, k) = 9*Pochhammer(11-k, k-1) * [x^n] x^k/Product_{j=1..k} (1-j*x).
T(n, k) = 9*Pochhammer(11-k, k-1)*S2(n, k) where S2(n, k) = A048993(n, k) are the Stirling numbers of the 2nd kind.

cp's OEIS Frontend

A180592 Digital root of 2n.

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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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