A010734 -id:A010734 - cp's OEIS frontend

A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5

Offset: 0

Paul Curtz, Dec 19 2008

Digital root of 2^n.
A regular version of Pitoun's sequence: a(n) = A029898(n+1).
Also obtained from permutations of A141425, A020806, A070366, A153110, A153990, A154127, A154687, or A154815.
This sequence and its (again period 6) repeated differences produce the table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4, -1, -2, -4,  1,  2,  4, -1, -2, ...
    1,  2, -5, -1, -2,  5,  1,  2, -5, -1, -2, ...
    1, -7,  4, -1,  7, -4,  1, -7,  4, -1,  7, ...
   -8, 11, -5,  8,-11,  5, -8, 11, -5,  8,-11, ...
   19,-16, 13,-19, 16,-13, 19,-16, 13,-19, 16, ...
  -35, 29,-32, 35,-29, 32,-35, 29,-32, 35,-29, ...
   64,-61, 67,-64, 61,-67, 64,-61, 67,-64, 61, ...
If each entry of this table is read modulo 9 we obtain the very regular table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
Also the decimal expansion of the constant 125/1001. - R. J. Mathar, Jan 23 2009
Digital root of the powers of any number congruent to 2 mod 9. - Alonso del Arte, Jan 26 2014

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

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

Cf. A004000, A007953, A010734, A010888, A029898, A030132, A082365, A145389, A189510.
Cf. digital roots of powers of c mod 9: c = 4, A100402; c = 5, A070366; c = 7, A070403; c = 8, A010689.
Cf. A020806, A070366, A141425, A153110, A153990, A154127, A154687, A154815.

&cat [[1, 2, 4, 8, 7, 5]^^30]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([1, 2, 4, 8, 7, 5]), n=0..40); # Wesley Ivan Hurt, Jul 05 2016

Flatten[Table[{1, 2, 4, 8, 7, 5}, {20}]] (* Paul Curtz, Dec 19 2008 *)
Table[Mod[2^n, 9], {n, 0, 99}] (* Alonso del Arte, Jan 26 2014 *)

a(n)=lift(Mod(2,9)^n) \\ Charles R Greathouse IV, Apr 21 2015

a(n) + a(n+3) = 9 = A010734(n).
G.f.: (1+x+2x^2+5x^3)/((1-x)(1+x)(1-x+x^2)). - R. J. Mathar, Jan 23 2009
a(n) = A082365(n) mod 9. - Paul Curtz, Mar 31 2009
a(n) = -1/2*cos(Pi*n) - 3*cos(1/3*Pi*n) - 3^(1/2)*sin(1/3*Pi*n) + 9/2. - Leonid Bedratyuk, May 13 2012
a(n) = A010888(A004000(n+1)). - Ivan N. Ianakiev, Nov 27 2014
From Wesley Ivan Hurt, Apr 20 2015: (Start)
a(n) = a(n-6) for n>5.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (2+3*(n-1 mod 3))*(n mod 2) + (1+3*(-n mod 3))*(n-1 mod 2). (End)
a(n) = 2^n mod 9. - Nikita Sadkov, Oct 06 2018
From Stefano Spezia, Mar 20 2025: (Start)
E.g.f.: 4*cosh(x) - exp(x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 5*sinh(x).
a(n) = A007953(2*a(n-1)) = A010888(2*a(n-1)). (End)

Edited by R. J. Mathar, Apr 09 2009

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 3, 1, 1, 0, 0, 3, 2, 1, 0, 5, 2, 3, 3, 1, 0, 0, 5, 4, 3, 4, 1, 0, 7, 3, 5, 6, 3, 5, 1, 0, 0, 7, 6, 5, 8, 3, 6, 1, 0, 9, 4, 7, 9, 5, 10, 3, 7, 1, 0, 0, 9, 8, 7, 12, 5, 12, 3, 8, 1, 0, 11, 5, 9, 12, 7, 15, 5, 14, 3, 9, 1, 0, 0, 11, 10, 9, 16, 7

Offset: 0

Omar E. Pol, Sep 14 2011

Also square array T(n,k) read by antidiagonals in which column k lists the multiples of k and the odd numbers interleaved, n>=0, k>=0. Also square array T(n,k) read by antidiagonals in which if n is even then row n lists the multiples of (n/2), otherwise if n is odd then row n lists a constant sequence: the all n's sequence. Partial sums of the numbers of column k give the column k of A195152. Note that if k >= 1 then partial sums of the numbers of the column k give the generalized m-gonal numbers, where m = k + 4.

All columns are multiplicative. - Andrew Howroyd, Jul 23 2018

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,...
.  3,   3,   3,   3,   3,   3,   3,   3,   3,   3,...
.  0,   2,   4,   6,   8,  10,  12,  14,  16,  18,...
.  5,   5,   5,   5,   5,   5,   5,   5,   5,   5,...
.  0,   3,   6,   9,  12,  15,  18,  21,  24,  27,...
.  7,   7,   7,   7,   7,   7,   7,   7,   7,   7,...
.  0,   4,   8,  12,  16,  20,  24,  28,  32,  36,...
.  9,   9,   9,   9,   9,   9,   9,   9,   9,   9,...
.  0,   5,  10,  15,  20,  25,  30,  35,  40,  45,...
...

Rows: A000004, A000012, A001477, A010701, A005843, A010716, A008585, A010727, A008586, A010734, A008587.

Columns k: A026741 (k=1), A001477 (k=2), zero together with A080512 (k=3), A022998 (k=4), A195140 (k=5), zero together with A165998 (k=6), A195159 (k=7), A195161 (k=8), A195312 k=(9), A195817 (k=10), A317311 (k=11), A317312 (k=12), A317313 (k=13), A317314 k=(14), A317315 (k=15), A317316 (k=16), A317317 (k=17), A317318 (k=18), A317319 k=(19), A317320 (k=20), A317321 (k=21), A317322 (k=22), A317323 (k=23), A317324 k=(24), A317325 (k=25), A317326 (k=26).

T(n,k) = n*((k-2)*(-1)^n+k+2)/4 \\ Andrew Howroyd, Jul 23 2018

A176522 Decimal expansion of (9+sqrt(85))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 23 2010

Continued fraction expansion of (9+sqrt(85))/2 is A010734.

			(9+sqrt(85))/2 = 9.10977222864644365500...

Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A010536 (decimal expansion of sqrt(85)), A010734 (all 9's sequence), A333345, A049310.

First[RealDigits[(9 + Sqrt[85])/2, 10, 100]] (* Paolo Xausa, Jun 18 2024 *)

(9+sqrt(85))/2 \\ Charles R Greathouse IV, Jul 24 2013

Equals lim_{n->infinity} S(n, sqrt(5*17))/S(n-1, sqrt(5*17)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A010680 Decimal expansion of 1/11.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Period 2: repeat [0,9].

			1/11 = 0.0909090909090909090909090909090909090909090909090909090909...

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

Cf. A000035, A010674.

Bisections give: A000004, A010734.

RealDigits[1/11, 10, 100][[1]] (* Alonso del Arte, Mar 11 2018 *)

a(n) = 9*(n%2); \\ Altug Alkan, Mar 25 2018

a(n) = (9/2)*(1 - (-1)^n) = 9*(n mod 2). - Paolo P. Lava, Oct 31 2006
From Elmo R. Oliveira, Jan 15 2024: (Start)
a(n) = a(n-2) for n >= 2.
a(n) = 3 * A010674(n).
G.f.: 9*x/(1-x^2).
E.g.f.: 9*sinh(x). (End)
a(n) = 9 * A000035(n). - Alois P. Heinz, Jan 16 2024

A021069 Decimal expansion of 1/65.

Original entry on oeis.org

0, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5

Offset: 0

N. J. A. Sloane

Without the leading 0 also the decimal expansion of 2/13.

			0.0153846153846...  - _Natan Arie Consigli_, Sep 18 2016

G. C. Greubel, Table of n, a(n) for n = 0..10000
Christian Táfula, Knights are 24/13 times faster than the king, arXiv:2405.19589 [math.CO], 2024.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A010734, A020806, A021017, A021095, A128834, A130806, A144472.

Magma
```
1./65; // G. C. Greubel, Jan 15 2018
```

Digits:=100: evalf(1/65); # Wesley Ivan Hurt, Sep 20 2016

Join[ConstantArray[0, Abs@ Last@ #], First@ #] &@ RealDigits[N[1/65, 120]] (* Michael De Vlieger, Sep 06 2016 *)

1./65. /* Michel Marcus, Aug 22 2016 */

Equals 2 - 24/13. See Táfula link. - Michel Marcus, May 31 2024

G.f.: x*(1 + 4*x - 2*x^2 + 6*x^3)/((1 - x)*(1 + x)*(1 - x + x^2)). - Stefano Spezia, Apr 30 2025

A023008 Number of partitions of n into parts of 9 kinds.

Original entry on oeis.org

1, 9, 54, 255, 1035, 3753, 12483, 38709, 113265, 315445, 841842, 2164185, 5382276, 12994290, 30543210, 70066809, 157199805, 345552183, 745377215, 1579915080, 3294664578, 6766656315, 13700560491, 27370137195, 53991639855, 105242612526, 202837976145

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010734. - Alois P. Heinz, Oct 17 2008

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. 9th column of A144064. - Alois P. Heinz, Oct 17 2008

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*9, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^9,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

a(n) ~ 3^(5/2) * exp(Pi * sqrt(6*n)) / (256 * n^3). - Vaclav Kotesovec, Feb 28 2015
a(0) = 1, a(n) = (9/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(9*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A140657 Powers of 2 with 3 alternatingly added and subtracted.

Original entry on oeis.org

4, -1, 7, 5, 19, 29, 67, 125, 259, 509, 1027, 2045, 4099, 8189, 16387, 32765, 65539, 131069, 262147, 524285, 1048579, 2097149, 4194307, 8388605, 16777219, 33554429, 67108867, 134217725, 268435459, 536870909, 1073741827, 2147483645, 4294967299, 8589934589

Offset: 0

Paul Curtz, Jul 10 2008

Jean-François Alcover and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 101 terms from Jean-François Alcover)
Index entries for linear recurrences with constant coefficients, signature (1, 2).

Cf. A000079, A007283, A010734, A033999, A096045, A140683.

[2^n+3*(-1)^n: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

LinearRecurrence[{1,2},{4,-1},40] (* or *) Total/@Partition[Riffle[ Table[ 2^n, {n,0,40}],{3,-3}],2] (* Harvey P. Dale, Nov 13 2014 *)
CoefficientList[Series[(4 - 5 x) / ((1 + x) (1 - 2 x)), {x, 0, 50}], x] (* Vincenzo Librandi, Jan 14 2015 *)

a(2n) = A000079(2n+1) + 3, a(2n+1) = A000079(2n+2) - 3.
a(n+1) - 2*a(n) = -9*A033999(n) = (-1)^(n+1)*A010734.
a(n) + a(n+1) = 3^*2^n = A007283(n).
a(2n) + a(2n+1) = A096045(n) + 2.
a(-n) = -A140683(n)/2^n.
O.g.f.: (4-5*x)/((1-2*x)(1+x)). - R. J. Mathar, Jul 29 2008
a(n) = 2^n+3*(-1)^n. - R. J. Mathar , Jul 29 2008

Edited and extended by R. J. Mathar, Jul 29 2008

4 inserted as first term and formulas accordingly updated by Jean-François Alcover, Jan 14 2015

A271880 Decimal expansion of the probability that a random real number is evil.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Apr 16 2016

A real number is said to be evil if the cumulative sum of its digits following the decimal point 'hits' the value 666. It is amazing how close this value is to 1/5 (the difference is in A271881).

			0.19999999999999999999999999999999999999999999999999999999999999978337...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Evil Number.

Cf. A001019, A010734, A100061, A100062, A271881.

RealDigits[SeriesCoefficient[(1 - x^9)/(x^10 - 10 x + 9), {x, 0, 665}], 10, 120][[1]] (* Amiram Eldar, May 24 2023 *)

0.0 + Vec(Ser((1-x^9)/(x^10-10*x+9),x,666))[666]

Equals A100061(666)/A100062(666).

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.

A381487 Numbers which are a power of their digital root.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 81, 128, 256, 512, 729, 2401, 6561, 8192, 16384, 32768, 59049, 78125, 524288, 531441, 823543, 1048576, 2097152, 4782969, 33554432, 43046721, 67108864, 134217728, 282475249, 387420489, 1220703125, 2147483648, 3486784401, 4294967296

Offset: 1

Stefano Spezia, Feb 25 2025

			a(12) = 128 is a term since 128 = 2^7 = A010888(128)^7.

Michel Marcus, Table of n, a(n) for n = 1..3347

Cf. A010888, A023106, A381491, A381492.

Digital root of k^n: A000012 (1), A153130 (2), A100401 (3), A100402 (4), A070366 (5), A100403 (6), A070403 (7), A010689 (8), A010734 (9).

A010888[n_]:=n - 9*Floor[(n-1)/9]; kmax=5*10^6; a={0,1}; For[k=2, k<=kmax, k++, If[A010888[k]!=1, If[IntegerQ[Log[A010888[k],k]], AppendTo[a,k]]]]; a

isok(k) = if ((k==0) || (k==1), return(1)); my(d=(k-1)%9+1); if (d>1, d^logint(k, d) == k); \\ Michel Marcus, Feb 26 2025

lista(nn) = my(list = List()); listput(list, 0); listput(list, 1); for (n=2, 9, for (k=1, logint(nn, n), if ((n^k-1)%9+1 == n, listput(list, n^k)););); vecsort(Vec(list)); \\ Michel Marcus, Feb 27 2025

a(n) = A381491(n)^A381492(n).

cp's OEIS Frontend

A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

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A195151 Square array read by antidiagonals upwards: T(n,k) = n*((k-2)*(-1)^n+k+2)/4, n >= 0, k >= 0.

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A176522 Decimal expansion of (9+sqrt(85))/2.

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A010680 Decimal expansion of 1/11.

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A021069 Decimal expansion of 1/65.

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A023008 Number of partitions of n into parts of 9 kinds.

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A140657 Powers of 2 with 3 alternatingly added and subtracted.

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A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.