A001019 -id:A001019 - cp's OEIS frontend

A008570 Digits of powers of 9.

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

Offset: 0

N. J. A. Sloane

The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). - Amiram Eldar, Mar 23 2025

			Triangle begins:
  1;
  9;
  8, 1;
  7, 2, 9;
  6, 5, 6, 1;
  5, 9, 0, 4, 9;
  5, 3, 1, 4, 4, 1;
  4, 7, 8, 2, 9, 6, 9;
  ...

Seiichi Manyama, Rows n = 0..100, flattened
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.

Cf. A001019.

Cf. A000455, A008564, A008565, A008566, A008567, A008568, A008569, A008571, A008572, A008573, A010054.

Flatten[IntegerDigits/@(9^Range[0,20])] (* Harvey P. Dale, Nov 17 2024 *)

A096046 a(n) = B(2n,3)/B(2n) (see comment).

Original entry on oeis.org

1, 15, 141, 1275, 11481, 103335, 930021, 8370195, 75331761, 677985855, 6101872701, 54916854315, 494251688841, 4448265199575, 40034386796181, 360309481165635, 3242785330490721, 29185067974416495, 262665611769748461

Offset: 0

Benoit Cloitre, Jun 17 2004

nonn

B(n,p) = Sum_{i=0..n} p^i*Sum_{j=0..i} binomial(n,j)*B(j) where B(k) = k-th Bernoulli number.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A096045, A096047, A096048.

[(1/4)*(7*9^n-3): n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

A096046(n):=(1/4)*(7*9^n-3)$ makelist(A096046(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

a(n)=sum(i=0,2*n,3^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n)

a(n) = (1/4)*(7*9^n - 3).
a(n) = 10*a(n-1) - 9*a(n-2); a(0)=1, a(1)=15.
a(n) = 9*a(n-1) + 6. First differences = 14*A001019(n). - Paul Curtz, Jul 07 2008

A096053 a(n) = (3*9^n - 1)/2.

Original entry on oeis.org

1, 13, 121, 1093, 9841, 88573, 797161, 7174453, 64570081, 581130733, 5230176601, 47071589413, 423644304721, 3812798742493, 34315188682441, 308836698141973, 2779530283277761, 25015772549499853, 225141952945498681

Offset: 0

Benoit Cloitre, Jun 18 2004

Generalized NSW numbers. - Paul Barry, May 27 2005
Counts total area under elevated Schroeder paths of length 2n+2, where area under a horizontal step is weighted 3. Case r=4 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315. Fifth binomial transform of (1+8x)/(1-16x^2), A107906. - Paul Barry, May 27 2005
Primes in this sequence include: a(2) = 13, a(4) = 1093, a(7) = 797161. Semiprimes in this sequence include: a(3) = 121 = 11^2, a(5) = 9841 = 13 * 757, a(6) = 88573 = 23 * 3851, a(9) = 64570081 = 1871 * 34511, a(10) = 581130733 = 1597 * 363889, a(12) = 47071589413 = 47 * 1001523179, a(19) = 225141952945498681 = 13097927 * 17189128703.
Sum of divisors of 9^n. - Altug Alkan, Nov 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A083420, A096045, A096046, A096047, A096054.

Cf. A107903, A138894 ((5*9^n-1)/4).

[(3*9^n-1)/2: n in [0..20]]; // Vincenzo Librandi, Nov 01 2011

Table[(3*9^n - 1)/2, {n, 0, 18}] (* L. Edson Jeffery, Feb 13 2015 *)

a(n)=(3*9^n-1)/2 \\ Charles R Greathouse IV, Sep 28 2015

vector(30, n, n--; sigma(9^n)) \\ Altug Alkan, Nov 10 2015

From Paul Barry, May 27 2005: (Start)
G.f.: (1+3*x)/(1-10*x+9*x^2);
a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*4^k;
a(n) = ((1+sqrt(4))*(5+2*sqrt(4))^n+(1-sqrt(4))*(5-2*sqrt(4))^n)/2. (End)
a(n-1) = (-9^n/3)*B(2n,1/3)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k)=B(k,0) is the k-th Bernoulli number.
a(n) = 10*a(n-1) - 9*a(n-2).
a(n) = 9*a(n-1) + 4. - Vincenzo Librandi, Nov 01 2011
a(n) = A000203(A001019(n)). - Altug Alkan, Nov 10 2015
a(n) = A320030(3^n-1). - Nathan M Epstein, Jan 02 2019

Edited by N. J. A. Sloane, at the suggestion of Andrew S. Plewe, Jun 15 2007

A223789 T(n,k)=Number of nXk 0..2 arrays with rows, diagonals and antidiagonals unimodal.

Original entry on oeis.org

3, 9, 9, 22, 81, 27, 46, 484, 729, 81, 86, 2116, 8635, 6561, 243, 148, 7396, 62365, 151580, 59049, 729, 239, 21904, 334230, 1560013, 2703137, 531441, 2187, 367, 57121, 1455816, 11012718, 39387861, 48302789, 4782969, 6561, 541, 134689, 5425943

Offset: 1

R. H. Hardin Mar 27 2013

Table starts
.....3..........9............22..............46................86
.....9.........81...........484............2116..............7396
....27........729..........8635...........62365............334230
....81.......6561........151580.........1560013..........11012718
...243......59049.......2703137........39387861.........343454446
...729.....531441......48302789......1026135371.......11150023974
..2187....4782969.....862007289.....27088106846......377163884938
..6561...43046721...15379566078....715394830136....12972494260444
.19683..387420489..274427327200..18858304684055...446829906314726
.59049.3486784401.4896915028511.496722962933967.15355124632228358

			Some solutions for n=3 k=4
..2..2..2..1....1..2..0..0....1..1..2..2....1..2..1..1....0..0..0..0
..0..2..2..1....0..0..1..0....0..2..2..1....1..1..2..0....0..1..2..0
..2..1..0..0....0..1..0..0....0..2..0..0....2..2..2..2....0..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..126

Column 1 is A000244
Column 2 is A001019
Row 1 is A223718
Row 2 is A223719

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 9*a(n-1)
k=3: [order 15]
k=4: [order 80]
Empirical: rows n=1..5 are polynomials of order 4*n for k>0,0,1,8,15

A223975 T(n,k)=Number of nXk 0..2 arrays with rows and antidiagonals unimodal.

Original entry on oeis.org

3, 9, 9, 22, 81, 27, 46, 484, 729, 81, 86, 2116, 9515, 6561, 243, 148, 7396, 76092, 186004, 59049, 729, 239, 21904, 440628, 2558848, 3628696, 531441, 2187, 367, 57121, 2026448, 22935921, 84988435, 70779056, 4782969, 6561, 541, 134689, 7829639

Offset: 1

R. H. Hardin Mar 30 2013

Table starts
.....3..........9.............22...............46.................86
.....9.........81............484.............2116...............7396
....27........729...........9515............76092.............440628
....81.......6561.........186004..........2558848...........22935921
...243......59049........3628696.........84988435.........1140963027
...729.....531441.......70779056.......2809740785........55803232969
..2187....4782969.....1380511272......92756321858......2708281019793
..6561...43046721....26926081924....3060966419662....131014406127439
.19683..387420489...525177301935..100999995564503...6329626912147424
.59049.3486784401.10243271456697.3332485315028073.305632588672082728

			Some solutions for n=3 k=4
..0..2..0..0....1..2..2..1....2..1..1..0....1..2..2..0....0..0..0..0
..0..1..2..1....0..0..1..1....1..2..2..2....0..2..1..0....0..2..2..0
..0..1..2..2....0..1..0..0....0..1..1..2....1..1..0..0....0..2..1..1

R. H. Hardin, Table of n, a(n) for n = 1..286

Column 1 is A000244
Column 2 is A001019
Row 1 is A223718
Row 2 is A223719

Empirical: columns k=1..6 have recurrences of order 1,1,7,18,43,91

Empirical: rows n=1..7 are polynomials of degree 4*n for k>0,0,0,0,2,3,4

A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ).

Original entry on oeis.org

1, 1, 10, 28, 136, 496, 2080, 8128, 32896, 130816, 524800, 2096128, 8390656, 33550336, 134225920, 536854528, 2147516416, 8589869056, 34359869440, 137438691328, 549756338176, 2199022206976, 8796095119360, 35184367894528, 140737496743936, 562949936644096, 2251799847239680

Offset: 0

N. J. A. Sloane

Binomial transform of expansion of cosh(3*x), the aerated version of A001019, 1,0,9,0,81,0,729,... - Paul Barry, Apr 05 2003
Alternatively: start with the fraction 1/1, take the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 9 times the bottom to get the new top. The limit of the sequence of fractions used to generate this sequence is sqrt(9). - Cino Hilliard, Sep 25 2005
This sequence also gives the number of ordered pairs of subsets (A, B) of {1, 2, ..., n} such that |A u B| is even. (Here "u" stands for the set-theoretic union.) The special case n = 13 can be found as in CRUX Problem 3257. - Walther Janous (walther.janous(AT)tirol.com), Mar 01 2008
For n > 0, a(n) is term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 3,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 9 types of other natural numbers. - Milan Janjic, Aug 13 2010
a(n) = ((1+3)^n+(1-3)^n)/2. In general, if b(0),b(1),... is the k-th binomial transform of the sequence ((3^n+(-3)^n)/2), then b(n) = ((k+3)^n+(k-3)^n)/2. More generally, if b(0),b(1),... is the k-th binomial transform of the sequence ((m^n+(-m)^n)/2), then b(n) = ((k+m)^n+(k-m)^n)/2. See A034494, A081340-A081342, A034659. - Charlie Marion, Jun 25 2011
Pisano period lengths: 1, 1, 1, 1, 4, 1, 6, 1, 1, 4, 5, 1, 12, 6, 4, 1, 8, 1, 9, 4, ... - R. J. Mathar, Aug 10 2012
Starting with offset 1 the sequence is the INVERT transform of (1, 9, 9, 9, ...). - Gary W. Adamson, Aug 06 2016

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.
M. Gardner, Riddles of Sphinx, M.A.A., 1987, p. 145.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Bill Sands, Problem 3257, Crux Math. 33 (2007), No.5, p. 298.
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001019, A001045, A014551, A078008, A098158.

Cf. A034494, A081340-A081342, A034659.

List([0..30], n-> 2^(n-1)*(2^n +(-1)^n)); # G. C. Greubel, Aug 02 2019

[2^(n-1)*( 2^n + (-1)^n ): n in [0..30]]; // Vincenzo Librandi, Aug 19 2011

A003665:=n->2^(n-1)*( 2^n + (-1)^n ): seq(A003665(n), n=0..30); # Wesley Ivan Hurt, Apr 28 2017

CoefficientList[Series[(1+8x)/(1-2x-8x^2), {x,0,30}], x] (* or *)
LinearRecurrence[{2,8}, {1,1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

PARI
```
a(n)=2^(n-1)*( 2^n + (-1)^n );
```

[2^(n-1)*(2^n +(-1)^n) for n in (0..30)] # G. C. Greubel, Aug 02 2019

From Paul Barry, Mar 01 2003: (Start)
a(n) = 2*a(n-1) + 8*a(n-2), a(0)=a(1)=1.
a(n) = (4^n + (-2)^n)/2.
G.f.: (1-x)/((1+2*x)*(1-4*x)). (End)
From Paul Barry, Apr 05 2003: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*9^k.
E.g.f. exp(x)*cosh(3*x). (End)
a(n) = (A078008(n) + A001045(n+1))*2^(n-1) = A014551(n)*2^(n-1). - Paul Barry, Feb 12 2004
Given a(0)=1, b(0)=1 then for i=1, 2, ..., a(i)/b(i) = (a(i-1) + 9*b(i-1)) / (a(i-1) + b(i-1)). - Cino Hilliard, Sep 25 2005
a(n) = Sum_{k=0..n} A098158(n,k)*9^(n-k). - Philippe Deléham, Dec 26 2007
a(n) = ((1+sqrt(9))^n + (1-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
If p[1]=1, and p[i]=9, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Apr 29 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(9*k-1)/(x*(9*k+8) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

Entry revised by N. J. A. Sloane, Nov 22 2006

A067403 Third column of triangle A067402.

Original entry on oeis.org

1, 5, 45, 405, 3645, 32805, 295245, 2657205, 23914845, 215233605, 1937102445, 17433922005, 156905298045, 1412147682405, 12709329141645, 114383962274805, 1029455660473245, 9265100944259205, 83385908498332845, 750473176484995605, 6754258588364960445, 60788327295284644005

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A002001 (second column), A067404 (fourth column), A001019 (powers of 9).

Cf. A067402.

A067403:=n->5*9^(n-1): 1,seq(A067403(n), n=1..30); # Wesley Ivan Hurt, Apr 09 2017

Join[{1},NestList[9#&,5,30]] (* or *) CoefficientList[Series[ (1-4x)/ (1-9x),{x,0,30}],x] (* Harvey P. Dale, Apr 26 2011 *)

Vec((1-4*x)/(1-9*x) + O(x^30)) \\ Michel Marcus, Apr 09 2017

a(n) = A067402(n+2, 2).
a(n) = 5*9^(n-1) for n>=1, a(0) = 1.
G.f.: (1-4*x)/(1-9*x).
E.g.f.: (4 + 5*exp(9*x))/9. - Stefano Spezia, Sep 30 2022

A075504 Stirling2 triangle with scaled diagonals (powers of 9).

Original entry on oeis.org

1, 9, 1, 81, 27, 1, 729, 567, 54, 1, 6561, 10935, 2025, 90, 1, 59049, 203391, 65610, 5265, 135, 1, 531441, 3720087, 1974861, 255150, 11340, 189, 1, 4782969, 67493007, 57041334, 11160261, 765450, 21546, 252, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(9*z) - 1)*x/9) - 1.
Row sums give A075508(n), n >= 1. The columns (without leading zeros) give A001019 (powers of 9), A076008-A076013 for m=1..7.

			[1]; [9,1]; [81,27,1]; ...; p(3,x) = x(81 + 27*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*       1
*       9        1
*      81       27        1
*     729      567       54        1
*    6561    10935     2025       90      1
*   59049   203391    65610     5265    135     1
*  531441  3720087  1974861   255150  11340   189   1
* 4782969 67493007 57041334 11160261 765450 21546 252 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A075503, A075505.

Columns 2-7 are A076008-A076013.

Flatten[Table[9^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(9^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (9^(n-m)) * stirling2(n, m).
a(n, m) = Sum_{p=0..m-1} (A075513(m, p)*((p+1)*9)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 9m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-9k*x), m >= 1.
E.g.f. for m-th column: (((exp(9x) - 1)/9)^m)/m!, m >= 1.

A100062 Denominator of the probability that an integer n occurs in the cumulative sums of the decimal digits of a random real number between 0 and 1.

Original entry on oeis.org

9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, 205891132094649, 1853020188851841, 16677181699666569, 150094635296999121

Offset: 1

Eric W. Weisstein, Nov 01 2004

Essentially the same as A001019 = powers of 9.

Also number of n-digit positive integers with no identical adjacent digits. Hence the numerator (with A052268 as denominator) of the probability that an n-digit positive integer has this property (e.g., 9/9, 81/90, 729/900, ..., where A100062(n)/A052268(n) reduces to A001019(n-1)/A011557(n-1)). - Rick L. Shepherd, Jun 08 2008

			1/9, 10/81, 100/729, 1000/6561, 10000/59049, ...

Stanislav Sykora, Table of n, a(n) for n = 1..666; previous Table by Rick L. Shepherd went up to n=30.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Evil Number
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019, A011557 A052268, A100061, A100062, A271880, A271881.

[9^n : n in [1..30]]; // Wesley Ivan Hurt, Apr 18 2016

A100062:=n->9^n: seq(A100062(n), n=1..30); # Wesley Ivan Hurt, Apr 18 2016

9^Range[20] (* Harvey P. Dale, Dec 25 2012 *)

\\ The 'old' approach, using the generating function:
s = Vec(Ser((1-x^9)/(x^10-10*x+9),x,666));
a = vector(#s,n,denominator(s[n])) \\ Stanislav Sykora, Apr 16 2016

a(n) = 9^n. - Max Alekseyev, Mar 03 2007
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 9*a(n-1), n>1; a(1)=9.
G.f.: 9x/(1-9x). (End)
a(n) = A001019(n) for n>0. - Wesley Ivan Hurt, Apr 18 2016

More terms from Rick L. Shepherd, Jun 08 2008

A155988 a(n) = (2n + 1)9^n.

Original entry on oeis.org

1, 27, 405, 5103, 59049, 649539, 6908733, 71744535, 731794257, 7360989291, 73222472421, 721764371007, 7060738412025, 68630377364883, 663426981193869, 6382625094934119, 61149666232110753, 583701359488329915, 5553501505988967477, 52683216989246691471, 498464283821334080841

Offset: 0

Jaume Oliver Lafont, Feb 01 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David H. Bailey, A Compendium of BBP-Type Formulas for Mathematical Constants, 2017, page 14.
Xavier Gourdon and Pascal Sebah, Collection of formulas for log 2.
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A058962 for the similar (2n+1)4^n.

Cf. A001019, A005408, A060851, A096949, A096950, A154920, A164985, A165132.

[(2*n+1)*9^n: n in [0..20]]; // Vincenzo Librandi, Jun 08 2011

makelist((2*n+1)*9^n, n, 0, 20); /* Martin Ettl, Nov 11 2012 */

PARI
```
a(n)=(2*n+1)*9^n;
```

G.f.: (1 + 9*x)/(1 - 9*x)^2.
a(n) = 18*a(n-1) - 81*a(n-2) for n>=2.
Sum_{n>=0} 1/a(n) = (3/2)*log(2).
a(n) = A005408(n) * A001019(n).
a(n) = (2*n - 1)*3^(2*n-1)/3 = A060851(n)/3.
Sum_{n>=0} (-1)^n/a(n) = 3*arctan(1/3). - Amiram Eldar, Feb 26 2022
E.g.f.: exp(9*x)*(1 + 18*x). - Stefano Spezia, May 07 2023

cp's OEIS Frontend

A008570 Digits of powers of 9.

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A096046 a(n) = B(2n,3)/B(2n) (see comment).

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Magma

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A096053 a(n) = (3*9^n - 1)/2.

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A223789 T(n,k)=Number of nXk 0..2 arrays with rows, diagonals and antidiagonals unimodal.

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A223975 T(n,k)=Number of nXk 0..2 arrays with rows and antidiagonals unimodal.

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Author

Keywords

Comments

Examples

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Crossrefs

Formula

A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ).

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GAP

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A067403 Third column of triangle A067402.

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Maple

A155988 a(n) = (2n + 1)9^n.