A019464 -id:A019464 - cp's OEIS frontend

A030297 a(n) = n*(n + a(n-1)) with a(0)=0.

0, 1, 6, 27, 124, 645, 3906, 27391, 219192, 1972809, 19728190, 217010211, 2604122676, 33853594957, 473950329594, 7109254944135, 113748079106416, 1933717344809361, 34806912206568822, 661331331924807979

Offset: 0

N. J. A. Sloane, "Urkonsaud_admin" (miti(AT)tula.sitek.net)

Exponential convolution of factorials (A000142) and squares (A000290). - Vladimir Reshetnikov, Oct 07 2016

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A019461-A019464, A006183, A054096, A111063.

f := proc(n) options remember; if n <= 1 then n elif n = 2 then 6 else -n*(n-2)*f(n-3)+(n-3)*n*f(n-2)+3*n*f(n-1)/(n-1); fi; end;

a=0;lst={a};Do[a=(a+n)*n;AppendTo[lst, a], {n, 2*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2008 *)
RecurrenceTable[{a[0]==0,a[n]==n(n+a[n-1])},a[n],{n,20}] (* Harvey P. Dale, Oct 22 2011 *)
Round@Table[(2 E Gamma[n, 1] - 1) n, {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 07 2016 *)

a(n) = A019461(2n).
For n>=2, a(n) = floor(2*e*n! - n - 2). - Benoit Cloitre, Feb 16 2003
a(n) = sum_{k=0...n} (n! / k!) * k^2. - Ross La Haye, Sep 21 2004
E.g.f.: x*(1+x)*exp(x)/(1-x). - Vladeta Jovovic, Dec 01 2004

Better description from Henry Bottomley, May 15 2000

A019460 Add 1, multiply by 1, add 2, multiply by 2, etc., start with 2.

Original entry on oeis.org

2, 3, 3, 5, 10, 13, 39, 43, 172, 177, 885, 891, 5346, 5353, 37471, 37479, 299832, 299841, 2698569, 2698579, 26985790, 26985801, 296843811, 296843823, 3562125876, 3562125889, 46307636557, 46307636571, 648306911994, 648306912009, 9724603680135, 9724603680151, 155593658882416

Offset: 0

N. J. A. Sloane

After a(7) = 43, the next prime in the sequence is a(649) with 676 digits. - M. F. Hasler, Jan 12 2011

New York Times, Oct 13, 1996.

Ivan Panchenko, Table of n, a(n) for n = 0..200
Nick Hobson, Python program for this sequence

Cf. A019461 (same, but start with 0), A019463 (start with 1), A019462 (start with 3), A082448 (start with 4).

Cf. A082458, A019464, A019465, A019466 (similar, but first multiply, then add; starting with 0,1,2,3).

a[n_] := If[ OddQ@n, a[n - 1] + (n + 1)/2, a[n - 1]*n/2]; a[0] = 2; Table[ a@n, {n, 0, 28}] (* Robert G. Wilson v, Jul 21 2009 *)

A019460(n)=2*(A000522(n\2)+(n\2)!)-if(bittest(n,0),1,n\2+2)
/* For producing the terms in increasing order, the following 'hack' can be used M. F. Hasler, Jan 12 2011 */
lastn=0; an1=1; A000522(n)={ an1=if(n, n==lastn && return(an1); n==lastn+1||error(); an1*lastn=n)+1 }

l=[2]
for n in range(1, 101):
    l.append(l[n - 1] + ((n + 1)//2) if n%2 else l[n - 1]*(n//2))
print(l) # Indranil Ghosh, Jul 05 2017

a(2n) = 2*(A000522(n) + n!) - n - 2.
a(2n+1) = 2*(A000522(n) + n!) - 1.
Recursive: a(0) = 2, a(n) = (1 + floor((n-1)/2) - ceiling((n-1)/2))*(a(n-1) + (n+2)/2) + (ceiling((n-1)/2) - floor((n-1)/2))*(n/2)*a(n-1). - Wesley Ivan Hurt, Jan 12 2013

One more term from Robert G. Wilson v, Jul 21 2009
Formula provided by Nathaniel Johnston, Nov 11 2010
Formula double-checked and PARI code added by M. F. Hasler, Nov 12 2010
Edited by M. F. Hasler, Feb 25 2018

A019466 Multiply by 1, add 1, multiply by 2, add 2, etc.; start with 3.

Original entry on oeis.org

3, 3, 4, 8, 10, 30, 33, 132, 136, 680, 685, 4110, 4116, 28812, 28819, 230552, 230560, 2075040, 2075049, 20750490, 20750500, 228255500, 228255511, 2739066132, 2739066144, 35607859872, 35607859885, 498510038390, 498510038404, 7477650576060, 7477650576075, 119642409217200

Offset: 0

N. J. A. Sloane

nonn

Paolo Xausa, Table of n, a(n) for n = 0..800

Cf. A082458 (same, but start with 0), A019464 (start with 1), A019465 (start with 2).

Cf. A019460 .. A019463 & A082448 (similar, but first add, then multiply).

Module[{a = 3}, Join[{a}, Flatten[Array[{a *= #, a += #} &, 20]]]] (* Paolo Xausa, Oct 24 2024 *)

A019466(n,a=3)={for(i=2,n+1,if(bittest(i,0),a+=i\2,a*=i\2));a} \\ M. F. Hasler, Feb 25 2018

Edited by M. F. Hasler, Feb 25 2018

A082458 Multiply by 1, add 1, multiply by 2, add 2, etc., starting with 0.

Original entry on oeis.org

0, 0, 1, 2, 4, 12, 15, 60, 64, 320, 325, 1950, 1956, 13692, 13699, 109592, 109600, 986400, 986409, 9864090, 9864100, 108505100, 108505111, 1302061332, 1302061344, 16926797472, 16926797485, 236975164790, 236975164804, 3554627472060, 3554627472075, 56874039553200, 56874039553216

Offset: 0

Vladeta Jovovic, Apr 25 2003

nonn

Bisections: A007526 and A038154.

Paolo Xausa, Table of n, a(n) for n = 0..800

Cf. A019464 (same, but start with 1), A019465 (start with 2), A019466 (start with 3).

Cf. A019460 .. A019463 & A082448 (similar, but first add, then multiply).

Module[{a = 0}, Join[{a}, Flatten[Array[{a *= #, a += #} &, 20]]]] (* Paolo Xausa, Oct 24 2024 *)

a(n)=if(n<2,0,if(n%2,(n+1)/2*(floor(exp(1)*((n-1)/2)!)-1),floor(exp(1)*(n/2)!)-1))

A082458(n,a=0)={for(i=2,n+1,if(bittest(i,0),a+=i\2,a*=i\2));a} \\ M. F. Hasler, Feb 25 2018

For n>=2, a(2n)=floor(e*n!)-1, a(2*n+1)=(n+1)*(floor(e*n!)-1). - Benoit Cloitre, Apr 28 2003

Edited by M. F. Hasler, Feb 25 2018

A019463 Add 1, multiply by 1, add 2, multiply by 2, etc., start with 1.

Original entry on oeis.org

1, 2, 2, 4, 8, 11, 33, 37, 148, 153, 765, 771, 4626, 4633, 32431, 32439, 259512, 259521, 2335689, 2335699, 23356990, 23357001, 256927011, 256927023, 3083124276, 3083124289, 40080615757, 40080615771, 561128620794, 561128620809, 8416929312135, 8416929312151, 134670868994416

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..900
Nick Hobson, Python program for this sequence.

Cf. A019461 (same, but start with 0), A019460 (start with 2), A019462, (start with 3), A082448. (start with 4).
Cf. A082458, A019464, A019465, A019466 (similar, but first multiply, then add).
Cf. A019762.

a:= proc(n) option remember; `if`(n=0, 1, (t->
      `if`(n::odd, t+(n+1)/2, t*n/2))(a(n-1)))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Jan 16 2024

For[i=1;lst={1},i<15,i++,AppendTo[lst,i+Last[lst]];AppendTo[lst,i Last[lst]]];lst (* Harvey P. Dale, Feb 25 2012 *)
FoldList[If[OddQ[#2], #1 + (#2 + 1)/2, #1 * (#2/2)]&, 1, Range[32]] (* AnneMarie Torresen, Nov 26 2023 *)

A019463(n, a=1)={for(i=2, n+1, if(bittest(i, 0), a*=i\2, a+=i\2)); a} \\ M. F. Hasler, Feb 25 2018

Limit_{n->oo} a(2n)/n! = 1 + 2e = 1 + A019762. - Jon E. Schoenfield, Jan 16 2024

Edited by M. F. Hasler, Feb 25 2018

A082448 Add 1, multiply by 1, add 2, multiply by 2, etc.; start with 4.

Original entry on oeis.org

4, 5, 5, 7, 14, 17, 51, 55, 220, 225, 1125, 1131, 6786, 6793, 47551, 47559, 380472, 380481, 3424329, 3424339, 34243390, 34243401, 376677411, 376677423, 4520129076, 4520129089, 58761678157, 58761678171, 822663494394, 822663494409, 12339952416135, 12339952416151, 197439238658416

Offset: 0

N. J. A. Sloane, based on a suggestion of Nick MacDonald, Apr 25 2003

Nick Hobson, Python program for this sequence

Cf. A019461 (same, but start with 0), A019463 (start with 1), A019460 (start with 2), A019462 (start with 3).

Cf. A082458, A019464 .. A019466 (similar, but first multiply, then add).

k = 0; NestList[(k++; {Last@# + k, k(k + Last@#)}) &, {4}, 16] // Flatten

a=4; for(n=1,150,print(a,","); b=if(n%2-1,a*ceil(n/2),a+ceil(n/2)); a=b)

A082448(n,a=4)={for(i=2,n+1,if(bittest(i,0),a*=i\2,a+=i\2));a} \\ M. F. Hasler, Feb 25 2018

For n>=2, a(2n)=floor((2e+4)*n!)-n-2, a(2n+1)=floor((2e+4)*n!)-1.

More terms from Benoit Cloitre, Apr 26 2003

Edited by M. F. Hasler, Feb 25 2018

A019461 Add 1, multiply by 1, add 2, multiply by 2, etc.; start with 0.

Original entry on oeis.org

0, 1, 1, 3, 6, 9, 27, 31, 124, 129, 645, 651, 3906, 3913, 27391, 27399, 219192, 219201, 1972809, 1972819, 19728190, 19728201, 217010211, 217010223, 2604122676, 2604122689, 33853594957, 33853594971, 473950329594, 473950329609, 7109254944135, 7109254944151, 113748079106416

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..899
Nick Hobson, Python program for this sequence

Cf. A019463 (same, but start with 1), A019460 (start with 2), A019462 (start with 3), A082448 (start with 4).

Cf. A082458, A019464 .. A019466 (similar, but first multiply, then add).

A019461 := proc(n) option remember; if n = 0 then 0 elif n mod 2 = 1 then (n+1)/2+A019461(n-1) else (n/2)*A019461(n-1); fi; end;

Module[{a = 0}, Join[{a}, Flatten[Array[{a += #, a *= #} &, 20]]]] (* Paolo Xausa, Oct 24 2024 *)

A019461(n,a=0)={for(i=2,n+1,if(bittest(i,0),a*=i\2,a+=i\2));a} \\ M. F. Hasler, Feb 25 2018

Edited by M. F. Hasler, Feb 25 2018

A019462 Add 1, multiply by 1, add 2, multiply by 2, etc., start with 3.

Original entry on oeis.org

3, 4, 4, 6, 12, 15, 45, 49, 196, 201, 1005, 1011, 6066, 6073, 42511, 42519, 340152, 340161, 3061449, 3061459, 30614590, 30614601, 336760611, 336760623, 4041127476, 4041127489, 52534657357, 52534657371, 735485203194, 735485203209, 11032278048135, 11032278048151

Offset: 0

N. J. A. Sloane

nonn

Paolo Xausa, Table of n, a(n) for n = 0..800
Nick Hobson, Python program for this sequence

Cf. A019461 (same, but start with 0), A019463 (start with 1), A019460 (start with 2), A082448 (start with 4).

Cf. A082458, A019464 .. A019466 (similar, but first multiply, then add).

Module[{a = 3}, Flatten[{a, Array[{a += #, a *= #} &, 20]}]] (* Paolo Xausa, Oct 24 2024 *)

A019462(n, a=3)={for(i=2, n+1, if(bittest(i, 0), a*=i\2, a+=i\2)); a} \\ M. F. Hasler, Feb 25 2018

Edited by M. F. Hasler, Feb 25 2018

A082459 Multiply by 1, add 1, multiply by 2, add 2, etc.

Original entry on oeis.org

-1, -1, 0, 0, 2, 6, 9, 36, 40, 200, 205, 1230, 1236, 8652, 8659, 69272, 69280, 623520, 623529, 6235290, 6235300, 68588300, 68588311, 823059732, 823059744, 10699776672, 10699776685, 149796873590, 149796873604, 2246953104060, 2246953104075, 35951249665200, 35951249665216

Offset: 0

Vladeta Jovovic, Apr 25 2003

sign

Bisections: A038156 and A038157.

Robert Israel, Table of n, a(n) for n = 0..898

Cf. A019464 - A019466.

seq(op(simplify([exp(1)*GAMMA(k+1,1)-k!-1, exp(1)*GAMMA(k+2,1)-(k+1)!-k-2])),k=0..20); # Robert Israel, Jan 11 2018

FoldList[If[OddQ@ #2, #1 ((#2 + 1)/2), #1 + #2/2] &, -1, Range@ 31] (* Michael De Vlieger, Jan 11 2018 *)

From Robert Israel, Jan 11 2018: (Start)
a(2*k) = e*Gamma(k+1,1) - k! - 1.
a(2*k+1) = e*Gamma(k+2,1) - (k+1)! - k - 2. (End)

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

cp's OEIS Frontend

A030297 a(n) = n*(n + a(n-1)) with a(0)=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A019460 Add 1, multiply by 1, add 2, multiply by 2, etc., start with 2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A019466 Multiply by 1, add 1, multiply by 2, add 2, etc.; start with 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A082458 Multiply by 1, add 1, multiply by 2, add 2, etc., starting with 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A019463 Add 1, multiply by 1, add 2, multiply by 2, etc., start with 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A082448 Add 1, multiply by 1, add 2, multiply by 2, etc.; start with 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A019461 Add 1, multiply by 1, add 2, multiply by 2, etc.; start with 0.

Views

Author

Keywords

Links

Crossrefs