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.

Showing 1-4 of 4 results.

A064753 a(n) = n*7^n - 1.

Original entry on oeis.org

6, 97, 1028, 9603, 84034, 705893, 5764800, 46118407, 363182462, 2824752489, 21750594172, 166095446411, 1259557135290, 9495123019885, 71213422649144, 531726889113615, 3954718737782518, 29311444762388081, 216579008522089716, 1595845325952240019, 11729463145748964146
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Links

Crossrefs

For a(n)=n*k^n-1 cf. -A000012 (k=0), A001477 (k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), this sequence (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), A064757 (k=11), A064758 (k=12).
Cf. A036293.

Programs

  • Magma
    [ n*7^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011
  • Maple
    k:= 7; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021
  • Mathematica
    Table[n 7^n-1,{n,20}] (* or *) LinearRecurrence[{15,-63,49},{6,97,1028},20] (* Harvey P. Dale, Feb 12 2022 *)

Formula

From Alois P. Heinz, Feb 19 2021: (Start)
G.f.: (56*x^2-21*x+1)/((x-1)*(7*x-1)^2).
a(n) = A036293(n) - 1. (End)
From Elmo R. Oliveira, May 05 2025: (Start)
E.g.f.: 1 + exp(x)*(7*x*exp(6*x) - 1).
a(n) = 15*a(n-1) - 63*a(n-2) + 49*a(n-3) for n > 3. (End)

A064756 a(n) = n*10^n - 1.

Original entry on oeis.org

9, 199, 2999, 39999, 499999, 5999999, 69999999, 799999999, 8999999999, 99999999999, 1099999999999, 11999999999999, 129999999999999, 1399999999999999, 14999999999999999, 159999999999999999, 1699999999999999999, 17999999999999999999, 189999999999999999999, 1999999999999999999999
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Links

Crossrefs

Cf. for a(n) = n*k^n - 1: -A000012 (k=0), A001477 (k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), this sequence (k=10), A064757 (k=11), A064758 (k=12).
Cf. A064748, A126431.

Programs

  • Magma
    [ n*10^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011
  • Maple
    k:= 10; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021
  • Mathematica
    Array[# 10^# - 1 &, 18] (* Michael De Vlieger, Jan 14 2020 *)

Formula

From Elmo R. Oliveira, Sep 07 2024: (Start)
G.f.: x*(100*x^2 - 10*x - 9)/((x - 1)*(10*x - 1)^2).
E.g.f.: 1 + exp(x)*(10*x*exp(9*x) - 1).
a(n) = 21*a(n-1) - 120*a(n-2) + 100*a(n-3) for n > 3.
a(n) = A126431(n) - 1 = A064748(n) - 2. (End)

A064758 a(n) = n*12^n - 1.

Original entry on oeis.org

11, 287, 5183, 82943, 1244159, 17915903, 250822655, 3439853567, 46438023167, 619173642239, 8173092077567, 106993205379071, 1390911669927935, 17974858503684095, 231105323618795519, 2958148142320582655, 37716388814587428863, 479219999055934390271, 6070119988041835610111, 76675199848949502443519
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Links

Crossrefs

Cf. for a(n) = n*k^n - 1: -A000012(k=0), A001477(k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), A064757 (k=11), this sequence (k=12).
Cf. A064750.

Programs

  • Magma
    [n*12^n - 1: n in [1..30]]; // Vincenzo Librandi, Jun 21 2018
  • Mathematica
    CoefficientList[Series[(11 + 12 x - 144 x^2) / ((1 - 12 x)^2 (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 21 2018 *)
  • PARI
    a(n) = { n*12^n - 1 } \\ Harry J. Smith, Sep 24 2009

Formula

G.f.: x*(11 + 12*x - 144*x^2)/((1 - 12*x)^2*(1 - x)). - Vincenzo Librandi, Jun 21 2018
From Elmo R. Oliveira, Sep 07 2024: (Start)
E.g.f.: 1 + exp(x)*(12*x*exp(11*x) - 1).
a(n) = 25*a(n-1) - 168*a(n-2) + 144*a(n-3) for n > 3.
a(n) = A064750(n) - 2. (End)

A064749 a(n) = n*11^n + 1.

Original entry on oeis.org

1, 12, 243, 3994, 58565, 805256, 10629367, 136410198, 1714871049, 21221529220, 259374246011, 3138428376722, 37661140520653, 448795257871104, 5316497670165375, 62658722541234766, 735195677817154577, 8592599484487994108, 100078511642860166659, 1162022718519876379530
Offset: 0

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Links

Crossrefs

For a(n)=n*k^n+1: A000012 (k=0), A000027(n+1) (k=1), A002064 (k=2), A050914 (k=3), A050915 (k=4), A050916 (k=5), A050917 (k=6), A050919 (k=7), A064746 (k=8), A064747 (k=9), A064748 (k=10), this sequence (k=11), A064750 (k=12).
Cf. A064757.

Programs

  • Magma
    [n*11^n+1: n in [0..20]]; // Vincenzo Librandi, Sep 16 2011
  • Maple
    k:= 11; f:= gfun:-rectoproc({-1 - (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(0) = 1, a(1) = k+1}, a(n), remember): map(f, [$0..20]); # Georg Fischer, Feb 19 2021

Formula

a(n) = A064757(n) + 2 for n>=1. - Georg Fischer, Feb 19 2021
G.f.: -(110*x^2-11*x+1)/((x-1)*(11*x-1)^2). - Alois P. Heinz, Feb 19 2021
From Elmo R. Oliveira, May 03 2025: (Start)
E.g.f.: exp(x)*(1 + 11*x*exp(10*x)).
a(n) = 23*a(n-1) - 143*a(n-2) + 121*a(n-3). (End)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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