A090805 -id:A090805 - cp's OEIS frontend

A033540 a(n+1) = n*(a(n) + 1) for n >= 1, a(1) = 1.

1, 2, 6, 21, 88, 445, 2676, 18739, 149920, 1349289, 13492900, 148421911, 1781062944, 23153818285, 324153456004, 4862301840075, 77796829441216, 1322546100500689, 23805829809012420, 452310766371235999, 9046215327424720000, 189970521875919120021

Offset: 1

Antti Karttunen

Vincenzo Librandi, Table of n, a(n) for n = 1..200
David Carlson, Sophomores meet the traveling salesperson problem, Journal of Computing Sciences in Colleges, 33(3), 126-133, 2018. See proof of the formula by Benoit Cloitre.
N. J. A. Sloane and Brady Haran, A Sequence with a Mistake, Numberphile video (2021).

Cf. A007526, A090805.

a:=[1,2,6];; for n in [4..30] do a[n]:=(n+1)*a[n-1]-(2*n-3)*a[n-2] +(n-3)*a[n-3]; od; a; # G. C. Greubel, Oct 13 2019

I:=[1,2,6]; [n le 3 select I[n] else (n+1)*Self(n-1)-(2*n-3)*Self(n-2)+(n-3)*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 21 2014

seq(coeff(series( (1+x*exp(x))/(1-x), x, n+1)*n!, x, n), n = 0..30); # G. C. Greubel, Oct 13 2019
# second Maple program:
a:= proc(n) option remember;
      `if`(n=1, 1, (n-1)*(a(n-1)+1))
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, May 12 2021

FoldList[#1*#2 + #2 &, 1, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
nxt[{a_,n_}]:={n(a+1),n+1}; Transpose[NestList[nxt,{1,1},20]][[1]] (* Harvey P. Dale, Jun 20 2014 *)

my(x='x+O('x^30)); Vec(serlaplace( (1+x*exp(x))/(1-x) )) \\ G. C. Greubel, Oct 13 2019

[factorial(n)*( (1+x*exp(x))/(1-x) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Oct 13 2019

a(n) = n!*(1 +1/0! +1/1! +...+ 1/(n-1)!). - Jon Bentley (jlb(AT)research.bell-labs.com)
For n>=1, a(n+1) = floor((1+e)*n!) - 1. - Benoit Cloitre, Sep 07 2002
From Vladeta Jovovic, Feb 02 2003: (Start)
a(n) = n! + A007526(n).
E.g.f.: (1+x*exp(x))/(1-x). (End)
a(n) = (n+1)*a(n-1) - (2*n-3)*a(n-2) + (n-3)*a(n-3) for n>=4. - Jaume Oliver Lafont, Sep 11 2009
a(n) = n! + floor(e*n!) - 1, n>0. - Gary Detlefs, Jun 06 2010

A344262 a(0)=1; for n>0, a(n) = a(n-1)n+1 if n is even, (a(n-1)+1)n otherwise.

Original entry on oeis.org

1, 2, 5, 18, 73, 370, 2221, 15554, 124433, 1119906, 11199061, 123189682, 1478276185, 19217590418, 269046265853, 4035693987810, 64571103804961, 1097708764684354, 19758757764318373, 375416397522049106, 7508327950440982121, 157674886959260624562

Offset: 0

Amrit Awasthi, May 13 2021

nonn

			a(0) = 1;
a(1) = (a(0)+1)*1 =  (1+1)*1 =   2;
a(2) = (a(1)*2)+1 =  (2*2)+1 =   5;
a(3) = (a(2)+1)*3 =  (5+1)*3 =  18;
a(4) = (a(3)*4)+1 = (18*4)+1 =  73;
a(5) = (a(4)+1)*5 = (73+1)*5 = 370.

Cf. A033540, A073743, A090805, A137204, A155521, A344317.

a:= proc(n) a(n):= n*a(n-1) + n^(n mod 2) end: a(0):= 1:
seq(a(n), n=0..22);  # Alois P. Heinz, May 14 2021

a[1] = 1; a[n_] := a[n] = If[OddQ[n], (n - 1)*a[n - 1] + 1, (n - 1)*(a[n - 1] + 1)]; Array[a, 25] (* Amiram Eldar, May 13 2021 *)

E.g.f.: (x+1)*cosh(x)/(1-x). - Alois P. Heinz, May 14 2021
Lim_{n->infinity} a(n)/n! = 2*cosh(1) = A137204 = 2*A073743. - Amrit Awasthi, May 15 2021
a(n) = A344317(n) - A155521(n-1) for n > 0. - Alois P. Heinz, May 18 2021

A344229 a(n) = n*a(n-1) + n^signum(n mod 4), a(0) = 1.

Original entry on oeis.org

1, 2, 6, 21, 85, 430, 2586, 18109, 144873, 1303866, 13038670, 143425381, 1721104573, 22374359462, 313241032482, 4698615487245, 75177847795921, 1278023412530674, 23004421425552150, 437084007085490869, 8741680141709817381, 183575282975906165022

Offset: 0

Alois P. Heinz, May 12 2021

nonn

This sequence is one of many possible solutions to puzzle 16 on the Meerdaelquiz puzzle page, cf. the Delestinne link and A090805.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Hugo Delestinne, Meerdaelquiz [broken link]
N. J. A. Sloane and Brady Haran, A Sequence with a Mistake, Numberphile video (2021)

Cf. A000142, A000522, A010873, A033540, A053817, A057427, A090805, A166486, A277506.

a:= proc(n) a(n):= n*a(n-1) + n^signum(n mod 4) end: a(0):= 1:
seq(a(n), n=0..23);

cp's OEIS Frontend

A033540 a(n+1) = n*(a(n) + 1) for n >= 1, a(1) = 1.

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A344262 a(0)=1; for n>0, a(n) = a(n-1)n+1 if n is even, (a(n-1)+1)n otherwise.

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Maple

Mathematica

Formula

A344229 a(n) = n*a(n-1) + n^signum(n mod 4), a(0) = 1.

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Author

Keywords

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Maple

cp's OEIS Frontend

A033540 a(n+1) = n*(a(n) + 1) for n >= 1, a(1) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A344262 a(0)=1; for n>0, a(n) = a(n-1)*n+1 if n is even, (a(n-1)+1)*n otherwise.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A344229 a(n) = n*a(n-1) + n^signum(n mod 4), a(0) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A344262 a(0)=1; for n>0, a(n) = a(n-1)n+1 if n is even, (a(n-1)+1)n otherwise.