A173316 -id:A173316 - cp's OEIS frontend

A020543 a(0) = 1, a(1) = 1, a(n+1) = (n+1)*a(n) + n.

1, 1, 3, 11, 47, 239, 1439, 10079, 80639, 725759, 7257599, 79833599, 958003199, 12454041599, 174356582399, 2615348735999, 41845579775999, 711374856191999, 12804747411455999, 243290200817663999, 4865804016353279999

Offset: 0

Simon Plouffe

nonn

First Bernoulli polynomial evaluated at x=n! and multiplied by 2.
From Jaroslav Krizek, Jan 23 2010: (Start)
a(0) = 1, for n >= 1: a(n) = numbers m for which there is one iteration {floor(r/k)} for k = n, n-1, n-2, ... 1 with property r mod k = k-1 starting at r = m.
For n = 5: a(5) = 239;
floor(239/5) = 47, 239 mod 5 = 4;
floor( 47/4) = 11,  47 mod 4 = 3;
floor( 11/3) =  3,  11 mod 3 = 2;
floor(  3/2) =  1,   3 mod 2 = 1;
floor(  1/1) =  1,   1 mod 1 = 0. (End)
With offset 1, is the eigensequence of a triangle with the natural numbers (1, 2, 3, ...) as the right border, (1, 1, 2, 3, 4, ...) as the left border; and the rest zeros. - Gary W. Adamson, Aug 01 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
Index entries for sequences related to Bernoulli numbers.

Cf. A052898(n) - 2.

Cf. sequences of the type k*n!-1: A033312 (k=1), this sequence, A173323 (k=3), A173321 (k=4), A173317 (k=5), A173316 (k=6).

[2*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013

lst={1};s=1;Do[s+=(n+=s*n);AppendTo[lst, s], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)
FoldList[#1*#2 + #2 - 1 &, 1, Range[20]] (* Robert G. Wilson v, Jul 07 2012 *)
Table[2 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

E.g.f.: (-2 + exp(x) - x*exp(x))/(1-x). - Ralf Stephan, Feb 18 2004
a(n) = 2*n! - 1. - Gary W. Adamson, Jan 07 2008
a(0) = a(1) = 1, a(n) = a(n-1) * n + (n-1) for n >= 2. - Jaroslav Krizek, Jan 23 2010
a(n) ~ 2*sqrt(2*Pi*n)*n^n/exp(n). - Ilya Gutkovskiy, Aug 02 2016

Better description from Benoit Cloitre, Dec 29 2001

A173321 a(n) = 4*n! - 1.

Original entry on oeis.org

3, 3, 7, 23, 95, 479, 2879, 20159, 161279, 1451519, 14515199, 159667199, 1916006399, 24908083199, 348713164799, 5230697471999, 83691159551999, 1422749712383999, 25609494822911999, 486580401635327999

Offset: 0

Vincenzo Librandi, Feb 16 2010

From Bernard Schott, Jul 11 2019: (Start)
With this sequence, it is possible to prove that there are infinitely many prime numbers of the form 4*k+3.
Prove that:
1. Every prime factor of a(n) is > n, and,
2. All these prime factors are of the form 4*k+1 or 4*k+3.
3. There is at least one prime of the form 4*k+3 > n,
4. The set of prime numbers of the form 4*k+3 is infinite.
(End)
The smallest prime of the form 4*k + 3 that divides a(n) is A333924(n). - Bernard Schott, Oct 08 2021

Transmath, Term S, Spécialité, Programme 2002, Nathan, 2002, Exercice 82 p. 93.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!-1: A033312 (k=1), A020543 (k=2), A173323 (k=3), this sequence, A173317 (k=5), A173316 (k=6).

Cf. A002145 (primes of the form 4*k+3), A333924.

[4*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013

[3] cat [n eq 1 select n+2 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013

A173321:=n->4*n! - 1; seq(A173321(n), n=0..25); # Wesley Ivan Hurt, Jan 24 2014

Table[4 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

a(n) = n*a(n-1) + n - 1 for n > 0, a(0) = 3. - Vincenzo Librandi, Sep 30 2013

A173317 a(n) = 5*n! - 1.

Original entry on oeis.org

4, 4, 9, 29, 119, 599, 3599, 25199, 201599, 1814399, 18143999, 199583999, 2395007999, 31135103999, 435891455999, 6538371839999, 104613949439999, 1778437140479999, 32011868528639999, 608225502044159999

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!-1: A033312 (k=1), A020543 (k=2), A173323 (k=3), A173321 (k=4), this sequence, A173316 (k=6).

[5*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013

[4] cat [n eq 1 select n+3 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013

Table[5 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

a(n) = 5*A000142(n)-1.

a(0)=4, a(n) = n*a(n-1)+n-1. - Vincenzo Librandi, Sep 30 2013

a(16) corrected from Vincenzo Librandi, Sep 30 2013

A173323 a(n) = 3*n! - 1.

Original entry on oeis.org

2, 2, 5, 17, 71, 359, 2159, 15119, 120959, 1088639, 10886399, 119750399, 1437004799, 18681062399, 261534873599, 3923023103999, 62768369663999, 1067062284287999, 19207121117183999, 364935301226495999, 7298706024529919999, 153272826515128319999, 3372002183332823039999

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!-1: A033312 (k=1), A020543 (k=2), this sequence, A173321 (k=4), A173317 (k=5), A173316 (k=6).

[3*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013

[2] cat [n eq 1 select n+1 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013

Table[3 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)
3*(Range[0,20]!)-1 (* Harvey P. Dale, Oct 27 2015 *)

a(0)=2, a(n) = n*a(n-1)+n-1. - Vincenzo Librandi, Sep 30 2013
D-finite with recurrence a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Mar 07 2022
E.g.f.: 3/(1 - x) - exp(x). - Stefano Spezia, Oct 14 2024

A229828 a(n) = 7*n! - 1.

Original entry on oeis.org

6, 6, 13, 41, 167, 839, 5039, 35279, 282239, 2540159, 25401599, 279417599, 3353011199, 43589145599, 610248038399, 9153720575999, 146459529215999, 2489811996671999, 44816615940095999, 851515702861823999, 17030314057236479999, 357636595201966079999

Offset: 0

Vincenzo Librandi, Sep 30 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!-1: A033312 (k=1), A020543 (k=2), A173323 (k=3), A173321 (k=4), A173317 (k=5), A173316 (k=6).

Magma
```
[7*Factorial(n)-1: n in [0..25]];
```

[6] cat [n eq 1 select n+5 else n*Self(n-1)+n-1: n in [1..25] ];

Mathematica
```
Table[7 n! - 1, {n, 0, 25}]
```

a(0)=6, a(n) = n*a(n-1)+n-1.

cp's OEIS Frontend

A020543 a(0) = 1, a(1) = 1, a(n+1) = (n+1)*a(n) + n.

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A173321 a(n) = 4*n! - 1.

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A173317 a(n) = 5*n! - 1.

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Magma

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A173323 a(n) = 3*n! - 1.

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Magma

Magma

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A229828 a(n) = 7*n! - 1.

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Magma

Magma

Mathematica

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