A001019 -id:A001019 - cp's OEIS frontend

A001556 a(n) = 1^n + 2^n + ... + 9^n.

9, 45, 285, 2025, 15333, 120825, 978405, 8080425, 67731333, 574304985, 4914341925, 42364319625, 367428536133, 3202860761145, 28037802953445, 246324856379625, 2170706132009733, 19179318935377305, 169842891165484965, 1506994510201252425

Offset: 0

N. J. A. Sloane

nonn

Conjectures for o.g.f.s for this type of sequences appear in the PhD thesis by Simon Plouffe. See A001552 for the reference. These conjectures are proved in the link given in A196837. - Wolfdieter Lang, Oct 15 2011

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 369
Index entries for linear recurrences with constant coefficients, signature (45, -870, 9450, -63273, 269325, -723680, 1172700, -1026576, 362880).

Column 9 of array A103438. A196837.

Table[Total[Range[9]^n], {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *)

a(n) = sum_{j=1..9} j^n, n>=0.
From Wolfdieter Lang, Oct 15 2011: (Start)
E.g.f.: (1-exp(9*x))/(exp(-x)-1) = sum(exp(j*x),j=1..9) (trivial).
O.g.f.: (9 - 360*x + 6090*x^2 - 56700*x^3 + 316365*x^4 - 1077300*x^5 + 2171040*x^6 - 2345400*x^7 + 1026576*x^8)/product_{j=1..9} (1-j*x).
From the e.g.f. via Laplace transformation. See the proof in a link under A196837.
(End)
a(n) = A001555(n) + A001019(n). - Michel Marcus, Jul 26 2013

More terms from Jon E. Schoenfield, Mar 24 2010

A009989 Powers of 45.

Original entry on oeis.org

1, 45, 2025, 91125, 4100625, 184528125, 8303765625, 373669453125, 16815125390625, 756680642578125, 34050628916015625, 1532278301220703125, 68952523554931640625, 3102863559971923828125, 139628860198736572265625, 6283298708943145751953125, 282748441902441558837890625

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 45), L(1, 45), P(1, 45), T(1, 45). Essentially same as Pisot sequences E(45, 2025), L(45, 2025), P(45, 2025), T(45, 2025). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 45-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (45).

Cf. A000244, A000351, A001019, A001024, A008776.

[45^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

45^Range[0,20] (* Harvey P. Dale, May 09 2012 *)

a(n)=45^n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 1/(1-45*x). - Philippe Deléham, Nov 24 2008
a(n) = 45^n; a(n) = 45*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(45*x).
a(n) = A000244(n)*A001024(n) = A000351(n)*A001019(n). (End)

A165293 Inverse of A038303, and generalization of A130595.

Original entry on oeis.org

1, 10, -1, 100, -20, 1, 1000, -300, 30, -1, 10000, -4000, 600, -40, 1, 100000, -50000, 10000, -1000, 50, -1, 1000000, -600000, 150000, -20000, 1500, -60, 1, 10000000, -7000000, 2100000, -350000, 35000, -2100, 70

Offset: 1

Mark Dols, Sep 13 2009

Rows sum up to A001019 (powers of 9), diagonals to A004189, a generalization of A010892 (the inverse Fibonacci). Ratio of diagonal sums converges to a decimal sequence: A000108 (Catalan numbers), which is the squared difference of sqrt(2) and sqrt(3), or 5-sqrt(24). Ratio between first binomial transform (A054320 and A138288)of A004189, converges to sqrt(2/3). 1/(2*sqrt(24)) gives A000984 (central binomial coefficients) as a decimal sequence.

Triangle T(n,k), read by rows, given by [10,0,0,0,0,0,0,0,...] DELTA [ -1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2009

			Triangle begins:
      1;
     10,    -1;
    100,   -20,   1;
   1000,  -300,  30,  -1;
  10000, -4000, 600, -40, 1;

Cf. A007318, A130595, A038303, A004189, A010892, A001079, A054320, A138288, A041041, A000108.

Sum_{k=0..n} T(n,k)*x^k = (10-x)^n. - Philippe Deléham, Dec 15 2009

G.f.: x*y/(1-10*x+x*y). - R. J. Mathar, Aug 11 2015

A270369 Expansion of g.f. (1-7x)/(1-9x).

Original entry on oeis.org

1, 2, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442, 774840978, 6973568802, 62762119218, 564859072962, 5083731656658, 45753584909922, 411782264189298, 3706040377703682, 33354363399333138, 300189270593998242, 2701703435345984178, 24315330918113857602, 218837978263024718418

Offset: 0

Colin Barker, Mar 18 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A054879 (partial sums), A132025.

Cf. similar sequences with g.f. (1-k*x)/(1-9*x) and k=0..8: A001019 (k=0; k=8 gives two initial 1's ), A055275 (k=1), A270472 (k=2), A092810 (k=3), A067403 (k=4), A270473 (k=5), A102518 (k=6), this sequence (k=7).

CoefficientList[Series[(1-7x)/(1-9x),{x,0,20}],x] (* or *) Join[ {1}, NestList[9#&,2,20]] (* Harvey P. Dale, Oct 15 2017 *)

PARI
```
Vec((1-7*x)/(1-9*x) + O(x^30))
```

G.f.: (1-7*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 2*9^(n-1) for n>0.
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 25/16.
Sum_{n>=0} (-1)^n/a(n) = 11/20.
Product_{n>=1} (1 - 1/a(n)) = A132025. (End)
E.g.f.: (2*exp(9*x) + 7)/9. - Elmo R. Oliveira, Mar 25 2025

A270473 Expansion of g.f. (1-5x)/(1-9x).

Original entry on oeis.org

1, 4, 36, 324, 2916, 26244, 236196, 2125764, 19131876, 172186884, 1549681956, 13947137604, 125524238436, 1129718145924, 10167463313316, 91507169819844, 823564528378596, 7412080755407364, 66708726798666276, 600378541187996484, 5403406870691968356, 48630661836227715204

Offset: 0

Colin Barker, Mar 17 2016

Also squares that can be expressed as the sum of two powers of three (3^x + 3^y), except a(0). - Karl-Heinz Hofmann, Sep 03 2022

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A083884 (partial sums).
Cf. A067403: (1-4*x)/(1-9*x); A102518: (1-6*x)/(1-9*x).
Cf. A356879, A356880.

Join[{1},NestList[9#&,4,20]] (* Harvey P. Dale, Oct 23 2022 *)

PARI
```
Vec((1-5*x)/(1-9*x) + O(x^30))
```

G.f.: (1-5*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 4*9^(n-1) for n>0.
E.g.f.: (4*exp(9*x) + 5)/9. - Stefano Spezia, Jul 09 2024

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).

A363249 Leading digit of 9^n.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 15 2023

He, Xinwei; Hildebrand, A J; Li, Yuchen; Zhang, Yunyi, Complexity of Leading Digit Sequences, Discrete Mathematics and Theoretical Computer Science; 22 (2020), #14.

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

Cf. A008952, A060956, A111395, A362871, A363093, A364185.

Cf. A000030, A001019, A008570.

a[n_] := IntegerDigits[9^n][[1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 15 2023 *)

PARI
```
a(n) = digits(9^n)[1];
```

a(n) = A000030(A001019(n)).

a(n) = A060956(2*n).

A013714 a(n) = 9^(2*n + 1).

Original entry on oeis.org

9, 729, 59049, 4782969, 387420489, 31381059609, 2541865828329, 205891132094649, 16677181699666569, 1350851717672992089, 109418989131512359209, 8862938119652501095929, 717897987691852588770249, 58149737003040059690390169, 4710128697246244834921603689

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (81).

Cf. A001019, A005408, A013778.

[9^(2*n+1): n in [0..20]]; // Vincenzo Librandi, May 26 2011

9^(2*Range[0,20]+1) (* or *) NestList[81#&,9,20] (* Harvey P. Dale, May 25 2020 *)

a(n)=9^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 81*a(n-1); a(0)=9.
G.f.: 9/(1-81*x). (End)
From Elmo R. Oliveira, Aug 26 2024: (Start)
E.g.f.: 9*exp(81*x).
a(n) = 3*A013778(n) = A001019(A005408(n)). (End)

A018870 9^a(n) is smallest power of 9 beginning with n.

Original entry on oeis.org

0, 12, 9, 7, 5, 4, 3, 2, 1, 21, 42, 20, 19, 40, 18, 17, 60, 16, 59, 15, 80, 14, 101, 57, 13, 78, 34, 12, 77, 33, 11, 98, 54, 10, 119, 75, 53, 9, 118, 96, 52, 30, 8, 95, 73, 51, 7, 138, 94, 72, 50, 28, 6, 115, 93, 71, 49, 27, 5, 114, 92, 70, 48, 26, 4, 135, 113, 91, 69, 47, 25, 3, 134

Offset: 1

David W. Wilson

			a(2) = 12 since 9^12 = 282429536481 is the smallest power of 9 that begins with 2.

D. Mondot, Table of n, a(n) for n = 1..32699

Cf. A001019, A018869.

def a(n):
    s, k = str(n), 0
    while not str(9**k).startswith(s): k += 1
    return k
print([a(n) for n in range(1, 74)]) # Michael S. Branicky, Dec 09 2021

A055692 Numbers k such that 9^k == -1 (mod k-1).

Original entry on oeis.org

2, 3, 11, 42, 147, 13818, 21450, 27594, 41370, 55011, 126291, 265722, 417123, 1315635, 3994571, 5704611, 6860490, 9298842, 13941354, 14349027, 17658578, 20382810, 26557874, 27841338, 69831363, 86550090, 113272170, 130457571, 163013235, 192688650, 211142538, 333792522

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

Cf. A001019.

Do[If[PowerMod[9, n, n-1]==n-2, Print[n]], {n, 2, 10^8}]

isok(k) = Mod(9, k-1)^k == -1; \\ Michel Marcus, Jul 12 2021

cp's OEIS Frontend

A001556 a(n) = 1^n + 2^n + ... + 9^n.

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A009989 Powers of 45.

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Magma

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A165293 Inverse of A038303, and generalization of A130595.

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A270369 Expansion of g.f. (1-7*x)/(1-9*x).

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Author

Keywords

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Crossrefs

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Mathematica

PARI

Formula

A270473 Expansion of g.f. (1-5*x)/(1-9*x).

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Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

Formula

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

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Formula

A363249 Leading digit of 9^n.

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Mathematica

PARI

Formula

A013714 a(n) = 9^(2*n + 1).

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A270369 Expansion of g.f. (1-7x)/(1-9x).

A270473 Expansion of g.f. (1-5x)/(1-9x).