cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A076008 Second column of triangle A075504.

Original entry on oeis.org

1, 27, 567, 10935, 203391, 3720087, 67493007, 1219657095, 21996874431, 396331160247, 7137447668847, 128505439098855, 2313380333315871, 41643387865514007, 749603858371707087, 13493075341822822215
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..1} (A075513(3,m)*exp(9*(m+1)*x)).

Links

Crossrefs

Cf. A001019, A076009.

Programs

  • Mathematica
    CoefficientList[Series[1/((1-9x)(1-18x)),{x,0,30}],x] (* or *) LinearRecurrence[{27,-162},{1,27},30] (* Harvey P. Dale, Dec 01 2015 *)

Formula

a(n) = A075504(n+2, 2) = (9^n)*S2(n+2, 2) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = -9^n + 2*18^n.
G.f.: 1/((1-9*x)*(1-18*x)).
E.g.f.: (d^2/dx^2)(((exp(9*x)-1)/9)^2)/2! = -exp(9*x) + 2*exp(18*x).
a(0)=1, a(1)=27, a(n) = 27*a(n-1) - 162*a(n-2). - Harvey P. Dale, Dec 01 2015

A104094 Largest prime <= 9^n.

Original entry on oeis.org

7, 79, 727, 6553, 59029, 531383, 4782961, 43046623, 387420479, 3486784393, 31381059607, 282429536453, 2541865828309, 22876792454939, 205891132094623, 1853020188851807, 16677181699666513, 150094635296999111
Offset: 1

Views

Author

Cino Hilliard, Mar 03 2005

Keywords

Links

Crossrefs

Cf. A013604.
Largest prime <= b^n: 2^n-A013603(n), 3^n-A013604(n), 4^n-A013606(n), 5^n-A013605(n), 6^n-A013607(n), 7^n-A013608(n), 8^n-A013603(3*n), 10^n-A033874(n).

Programs

  • Maple
    f:= n -> prevprime(9^n):
map(f, [$1..30]); # Robert Israel, Aug 12 2019
  • Mathematica
    NextPrime[#,-1]&/@(9^Range[20]) (* Harvey P. Dale, Apr 21 2024 *)
  • PARI
    g(n,b) = for(x=0,n,print1(precprime(b^x)","))

Formula

a(n) = 9^n - A013604(2*n) = A001019(n) - A013604(2*n), n > 0. A.H.M. Smeets, Aug 12 2019

A134007 a(n) = 1^n + 3^n + 5^n + 7^n + 9^n.

Original entry on oeis.org

5, 25, 165, 1225, 9669, 79225, 665445, 5686825, 49208709, 429746905, 3779084325, 33407391625, 296515495749, 2639977136185, 23561123826405, 210669225531625, 1886405750358789, 16910575282247065, 151726863979595685
Offset: 0

Views

Author

Artur Jasinski, Oct 01 2007

Keywords

Examples

			a(3)=165 because 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 165.

Links

Crossrefs

Cf. A034472, A074507, A134006.

Programs

  • Magma
    [1^n + 3^n + 5^n + 7^n + 9^n: n in [0..20]]; // Vincenzo Librandi, Jun 20 2011
  • Mathematica
    Table[1^n+3^n+5^n+7^n+9^n,{n,0,30}]

Formula

a(n) = 24*a(n-1) - 206*a(n-2) + 744*a(n-3) - 945*a(n-4) + 384.
G.f.: -(5 - 100*x + 690*x^2 - 1900*x^3 + 1689*x^4)/((-1+x)*(3*x-1)*(9*x-1)*(7*x-1)*(5*x-1)). - R. J. Mathar, Nov 14 2007
a(n) = A134006(n) + A001019(n). - Michel Marcus, Nov 07 2013

A144073 Euler transform of powers of 9.

Original entry on oeis.org

1, 9, 126, 1623, 20583, 254493, 3091803, 36974025, 436377771, 5091463423, 58811218362, 673298882775, 7647050353038, 86229872235432, 966019964324004, 10757807941399023, 119146632352548516, 1312935665205028374, 14400230629085596621, 157253909597473608945
Offset: 0

Views

Author

Alois P. Heinz, Sep 09 2008

Keywords

Links

Crossrefs

9th column of A144074.
Cf. A001019 (powers of 9).

Programs

  • Maple
    with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->9^j)(n): seq(a(n), n=0..40);
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1-x^j)^(9^j), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 14 2015 *)

Formula

G.f.: Product_{j>0} 1/(1-x^j)^(9^j).
a(n) ~ 9^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(9^(m-1)-1)) = 0.0670436814415340801450018457068097893307906... . - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(9*Sum_{k>=1} x^k/(k*(1 - 9*x^k))). - Ilya Gutkovskiy, Nov 10 2018

A158749 a(n) = n*9^n.

Original entry on oeis.org

0, 9, 162, 2187, 26244, 295245, 3188646, 33480783, 344373768, 3486784401, 34867844010, 345191655699, 3389154437772, 33044255768277, 320275094369454, 3088366981419735, 29648323021629456, 283512088894331673, 2701703435345984178, 25666182635786849691, 243153309181138576020
Offset: 0

Views

Author

Zerinvary Lajos, Mar 25 2009

Keywords

Links

Crossrefs

Cf. A001019, A018215, A036289, A036290, A036291, A036292, A036293, A036294.
Cf. A038299, A126431.

Programs

Formula

a(n) = n*9^n.
From R. J. Mathar, Mar 26 2009: (Start)
a(n) = 18*a(n-1) - 81*a(n-2) = A038299(n,1).
G.f.: 9*x/(1-9*x)^2. (End)
a(n) = A001019(n)*n. - Omar E. Pol, Mar 26 2009
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(9/8).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(10/9). (End)
E.g.f.: 9*x*exp(9*x). - Elmo R. Oliveira, Sep 09 2024

A165427 a(1) = 1, a(2) = 9, a(n) = product of the previous terms for n >= 3.

Original entry on oeis.org

1, 9, 9, 81, 6561, 43046721, 1853020188851841, 3433683820292512484657849089281, 11790184577738583171520872861412518665678211592275841109096961
Offset: 1

Views

Author

Jaroslav Krizek, Sep 17 2009

Keywords

Links

Programs

  • Mathematica
    a[1]:= 1; a[2]:= 9; a[n_]:= Product[a[j], {j,1,n-1}]; Table[a[n],{n,1, 12}] (* G. C. Greubel, Oct 19 2018 *)
  • PARI
    {a(n) = if(n==1, 1, if(n==2, 9, prod(j=1,n-1, a(j))))};
for(n=1,10, print1(a(n), ", ")) \\ G. C. Greubel, Oct 19 2018

Formula

a(1) = 1, a(2) = 9, a(n) = Product_{i = 1..n-1} a(i), n >= 3.
a(1) = 1, a(2) = 9, a(n) = A001019(2^(n-3)) = 9^(2^(n-3)), n >= 3.
a(1) = 1, a(2) = 9, a(3) = 9, a(n) = (a(n-1))^2, n >= 4.
a(n) = A011764(n-2), n > 2. - R. J. Mathar, Sep 20 2009

A177095 9^n - 8.

Original entry on oeis.org

1, 73, 721, 6553, 59041, 531433, 4782961, 43046713, 387420481, 3486784393, 31381059601, 282429536473, 2541865828321, 22876792454953, 205891132094641, 1853020188851833, 16677181699666561, 150094635296999113
Offset: 1

Views

Author

Vincenzo Librandi, Nov 15 2010

Keywords

Links

Crossrefs

Equals A001019(n)-8.

Programs

  • Magma
    [9^n-8: n in [1..20]];
  • Mathematica
    CoefficientList[Series[(1 + 63 x)/(1 - 10 x + 9 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)
LinearRecurrence[{10,-9},{1,73},20] (* Harvey P. Dale, May 22 2014 *)

Formula

G.f.: x*(1 + 63*x)/(1 - 10*x + 9*x^2). - Vincenzo Librandi, Feb 06 2013
a(n) = 10*a(n-1) - 9*a(n-2) for n>2, a(1)=1, a(2)=73. - Vincenzo Librandi, Feb 06 2013

A198478 a(n) = 9^n * n^9.

Original entry on oeis.org

0, 9, 41472, 14348907, 1719926784, 115330078125, 5355700839936, 193010051319183, 5777633090469888, 150094635296999121, 3486784401000000000, 73994897046174912819, 1457274373159131021312, 26955214582765006137717
Offset: 0

Views

Author

Vincenzo Librandi, Oct 27 2011

Keywords

Links

Crossrefs

Cf. A001017, A001019.
Cf. A007758, A062074, A062075, A097206, A098803, A098880.

Programs

  • Magma
    [9^n*n^9: n in [0..20]]
  • Mathematica
    Table[9^n*n^9, {n, 0, 20}] (* G. C. Greubel, May 17 2022 *)
  • SageMath
    [9^n*n^9 for n in (0..20)] # G. C. Greubel, May 17 2022

Formula

G.f.: 9*x*(1 + 4518*x + 1183248*x^2 + 64322586*x^3 + 1024762590*x^4 + 5210129466*x^5 + 7763290128*x^6 + 2401050438*x^7 + 43046721*x^8)/(1 - 9*x)^10. - Colin Barker, Apr 30 2013
a(n) = A001019(n)*A001017(n). - Michel Marcus, May 18 2022

A230540 a(n) = 2*n*3^(2*n-1).

Original entry on oeis.org

0, 6, 108, 1458, 17496, 196830, 2125764, 22320522, 229582512, 2324522934, 23245229340, 230127770466, 2259436291848, 22029503845518, 213516729579636, 2058911320946490, 19765548681086304, 189008059262887782, 1801135623563989452, 17110788423857899794
Offset: 0

Views

Author

Bruno Berselli, Oct 23 2013

Keywords

Comments

Arithmetic derivative of 9^n: a(n) = A003415(9^n).
Sum of reciprocals of a(n), for n>0: (3/2)*log(9/8).

Links

Crossrefs

Cf. A001019, A003415.
Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), A230539 (k=8), this sequence, A085708 (k=10), A081127 (k=11).

Programs

  • Magma
    [2*n*3^(2*n-1): n in [0..20]];
  • Mathematica
    Table[2 n 3^(2 n - 1), {n, 0, 20}]
  • PARI
    a(n) = 2*n*3^(2*n-1); \\ Michel Marcus, Oct 23 2013

Formula

G.f.: 6*x/(1-9*x)^2.
a(n) = 6*A053540(n), with A053540(0)=0.

A270472 Expansion of g.f. (1-2*x)/(1-9*x).

Original entry on oeis.org

1, 7, 63, 567, 5103, 45927, 413343, 3720087, 33480783, 301327047, 2711943423, 24407490807, 219667417263, 1977006755367, 17793060798303, 160137547184727, 1441237924662543, 12971141321962887, 116740271897665983, 1050662447078993847, 9455962023710944623, 85103658213398501607
Offset: 0

Views

Author

Colin Barker, Mar 17 2016

Keywords

Links

Crossrefs

Cf. A001019 (powers of 9), A005032, A187709 (partial sums).
Cf. A055275: (1-x)/(1-9*x); A092810: (1-3*x)/(1-9*x).

Programs

  • Mathematica
    CoefficientList[Series[(1 - 2 x)/(1 - 9 x), {x, 0, 20}], x] (* Michael De Vlieger, Mar 18 2016 *)
  • PARI
    Vec((1-2*x)/(1-9*x) + O(x^30))

Formula

G.f.: (1-2*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 7*9^(n-1) for n>0.
a(n) = A005032(2*n-2). - R. J. Mathar, Jan 28 2025
E.g.f.: (7*exp(9*x) + 2)/9. - Elmo R. Oliveira, Mar 25 2025
Previous Showing 61-70 of 117 results. Next

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