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 1-7 of 7 results.

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

Views

Author

Vincenzo Librandi, Feb 16 2010

Keywords

Comments

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

References

  • Transmath, Term S, SpĂ©cialitĂ©, Programme 2002, Nathan, 2002, Exercice 82 p. 93.

Links

Crossrefs

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.

Programs

Formula

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

A173316 a(n) = 6*n! - 1.

Original entry on oeis.org

5, 5, 11, 35, 143, 719, 4319, 30239, 241919, 2177279, 21772799, 239500799, 2874009599, 37362124799, 523069747199, 7846046207999, 125536739327999, 2134124568575999, 38414242234367999, 729870602452991999, 14597412049059839999, 306545653030256639999
Offset: 0

Views

Author

Vincenzo Librandi, Feb 16 2010

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [6*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013
  • Magma
    [5] cat [n eq 1 select n+4 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013
  • Mathematica
    Table[6 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

Formula

a(0)=5, a(n) = n*a(n-1)+n-1. - 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

Views

Author

Vincenzo Librandi, Feb 16 2010

Keywords

Links

Crossrefs

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).

Programs

  • Magma
    [5*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013
  • Magma
    [4] cat [n eq 1 select n+3 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013
  • Mathematica
    Table[5 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

Formula

a(n) = 5*A000142(n)-1.
a(0)=4, a(n) = n*a(n-1)+n-1. - Vincenzo Librandi, Sep 30 2013

Extensions

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

Views

Author

Vincenzo Librandi, Feb 16 2010

Keywords

Links

Crossrefs

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).

Programs

Formula

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

A136573 Triangle read by rows: (A000012 * A136572 + A136572 * A000012) - A000012.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 6, 6, 7, 11, 24, 24, 25, 29, 47, 120, 120, 121, 125, 143, 239, 720, 720, 721, 725, 743, 839, 1439, 5040, 5040, 5041, 5045, 5063, 5159, 5759, 10079, 40320, 40320, 40321, 40325, 40343, 40439, 41039, 45359, 80639, 362880, 362880, 362881, 362885, 362903, 362999, 363599, 367919, 403199, 725759
Offset: 0

Views

Author

Gary W. Adamson, Jan 07 2008

Keywords

Comments

Row sums = A136574.
Right border = 2*n! - 1 = A020543: (1, 1, 3, 11, 47, 239, 1439, ...).

Examples

			First few rows of the triangle:
     1;
     1,    1;
     2,    2,    3;
     6,    6,    7,   11;
    24,   24,   25,   49,   47;
   120,  120,  121,  125,  143,  239;
   720,  720,  721,  725,  743,  839, 1439;
  5040, 5040, 5041, 5045, 5063, 5159, 5759, 10079;  ...
Row 4 = (24, 24, 25, 29, 47) = 5 terms of (24, 24, 24, 24, 24) + (0, 0, 1, 5, 23), where A033312 = (0, 0, 1, 5, 23, 119, 719, 5039, ...).

Crossrefs

Cf. A020543, A033312, A136572, A136574.

Formula

(A000012 * A136572 + A136572 * A000012) - A000012, as infinite lower triangular matrices.
Triangle read by rows: n-th row = (n+1) terms of n! + (k! - 1), k = 0, 1, 2, ...; where the sequence (k! - 1) = A033312: (0, 0, 1, 5, 23, 119, 719, 5039, ...).

Extensions

a(41) corrected and more terms from Georg Fischer, Jun 05 2023

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

Views

Author

Vincenzo Librandi, Sep 30 2013

Keywords

Links

Crossrefs

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).

Programs

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

Formula

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

A350855 a(0) = 1, a(n) = (n+1)*a(n-1) + (n-2).

Original entry on oeis.org

1, 1, 3, 13, 67, 405, 2839, 22717, 204459, 2044597, 22490575, 269886909, 3508529827, 49119417589, 736791263847, 11788660221565, 200407223766619, 3607330027799157, 68539270528183999, 1370785410563679997, 28786493621837279955, 633302859680420159029, 14565965772649663657687
Offset: 0

Views

Author

Amrit Awasthi, Jan 19 2022

Keywords

Examples

			a(1) = (1+1)*a(0) + (1-2) = 2-1 = 1.
a(2) = (2+1)*a(1) + (2-2) = 3.

Links

Crossrefs

Cf. A020543.

Programs

  • Mathematica
    Nest[Append[#1, (#2 + 1) #1[[-1]] + (#2 - 2)] & @@ {#, Length@ #} &, {1}, 20] (* Michael De Vlieger, Jan 19 2022 *)
nxt[{n_,a_}]:={n+1,a(n+2)+n-1}; NestList[nxt,{0,1},30][[;;,2]] (* Harvey P. Dale, Feb 03 2025 *)
  • PARI
    a(n) = if (n, (n+1)*a(n-1) + (n-2), 1); \\ Michel Marcus, Jan 19 2022
  • Python
    terms = [1]
for n in range(1, 20):
    terms.append((n+1)*terms[-1]+n-2)
print(terms) # Gleb Ivanov, Jan 19 2022

Formula

a(n) ~ (6-2e)*(n+1)!.
E.g.f.: (exp(x)*(4*x-x^2-5)+6)/(x-1)^2. - Alois P. Heinz, Jan 19 2022
Showing 1-7 of 7 results.

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