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 1-10 of 11 results. Next

A019464 Multiply by 1, add 1, multiply by 2, add 2, etc., start with 1.

Original entry on oeis.org

1, 1, 2, 4, 6, 18, 21, 84, 88, 440, 445, 2670, 2676, 18732, 18739, 149912, 149920, 1349280, 1349289, 13492890, 13492900, 148421900, 148421911, 1781062932, 1781062944, 23153818272, 23153818285, 324153455990, 324153456004, 4862301840060, 4862301840075, 77796829441200
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A033540 (=a(2n)).
Cf. A082458 (same, but start with 0), A019465 (start with 2), A019466 (start with 3).
Cf. A019460 .. A019463 & A082448 (similar, but first add, then multiply).

Programs

  • Haskell
    a019464 n = a019464_list !! n
a019464_list = 1 : concat (unfoldr ma (1, [1, 1])) where
   ma (x, [_, j]) = Just (ij', (x + 1, ij')) where ij' = [x * j, x * j + x]
-- Reinhard Zumkeller, Nov 14 2011
  • Mathematica
    a[n_?EvenQ] := n/2 + a[n-1]; a[n_?OddQ] := (n+1)*a[n-1]/2;
a[0] = 1; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Nov 15 2011 *)
  • PARI
    A019464(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

Formula

For n>=1, a(2n)=floor((1+e)*(n-1)!)-1, a(2n+1)=floor((1+e)*(n+1)!)-n-2. - Benoit Cloitre, Apr 29 2003
a(n+1) = (1/2)*a(n)*(n+1 mod 2)*(n+2) + (1/2)*(n mod 2)*(2*a(n)+n+1). - Francois Jooste (pin(AT)myway.com), Jun 25 2003
a(n) = (n mod 2)*(floor((1+e)*(floor(n/2)+1)!)-floor(n/2)-2)+((n+1) mod 2)*(floor((1+e)*floor(n/2)!)-1) for n >= 1 with a(0) = 1. - Wesley Ivan Hurt, Aug 04 2025

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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).

Programs

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

Extensions

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

Views

Author

Vladeta Jovovic, Apr 25 2003

Keywords

Comments

Bisections: A007526 and A038154.

Links

Crossrefs

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).

Programs

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

Formula

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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

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

Views

Author

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

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    k = 0; NestList[(k++; {Last@# + k, k(k + Last@#)}) &, {4}, 16] // Flatten
  • PARI
    a=4; for(n=1,150,print(a,","); b=if(n%2-1,a*ceil(n/2),a+ceil(n/2)); a=b)
  • PARI
    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

Formula

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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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).

Programs

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

Extensions

Edited by M. F. Hasler, Feb 25 2018

A019465 Multiply by 1, add 1, multiply by 2, add 2, etc., start with 2.

Original entry on oeis.org

2, 2, 3, 6, 8, 24, 27, 108, 112, 560, 565, 3390, 3396, 23772, 23779, 190232, 190240, 1712160, 1712169, 17121690, 17121700, 188338700, 188338711, 2260064532, 2260064544, 29380839072, 29380839085, 411331747190, 411331747204, 6169976208060, 6169976208075, 98719619329200
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A082458 (same, but start with 0), A019465 (start with 2), A019466 (start with 3).
Cf. A019460 .. A019463 & A082448 (similar, but first add, then multiply).

Programs

  • Maple
    A[0]:= 2:
for n from 0 to 14 do
  A[2*n+1]:= (n+1)*A[2*n];
  A[2*n+2]:= (n+1)+A[2*n+1];
od:
seq(A[i],i=0..30); # Robert Israel, Dec 22 2015
  • Mathematica
    a = {2}; Do[If[EvenQ@ Length@ a, AppendTo[a, Floor[Length[a]/2] Last@ a],
AppendTo[a, Last@ a + Floor[Length[a] /2]]], {k, 27}]; Rest@ a (* Michael De Vlieger, Dec 22 2015 *)
  • PARI
    A019465(n,a=2)={for(i=2,n+1,if(bittest(i,0),a+=i\2,a*=i\2));a} \\ M. F. Hasler, Feb 25 2018

Formula

From Robert Israel, Dec 22 2015: (Start)
a(2*k) = 2*k! + Sum_{j=0..k-1} k!/j! = 2*k! + k*e*Gamma(k,1).
a(2*k+1) = 2*(k+1)! + Sum_{j=0..k-1} (k+1)!/j! = 2*(k+1)! + k*(k+1)*e*Gamma(k,1).
a(n) ~ (e+2)*(ceiling(n/2))!. (End)

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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).

Programs

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

Extensions

Edited by M. F. Hasler, Feb 25 2018

A053817 a(0)=1, a(n) = n*(a(n-1) + n).

Original entry on oeis.org

1, 2, 8, 33, 148, 765, 4626, 32431, 259512, 2335689, 23356990, 256927011, 3083124276, 40080615757, 561128620794, 8416929312135, 134670868994416, 2289404772905361, 41209285912296822, 782976432333639979, 15659528646672799980, 328850101580128800021
Offset: 0

Views

Author

mindy and steve schwartz (minsteve(AT)netvision.net.il), Apr 07 2000

Keywords

Examples

			Start with 1, add 1, multiply by 1 (getting 2), add 2, multiply by 2 (getting 8).

Crossrefs

Cf. A019460, A030297.

Programs

  • Maple
    a:= proc(n) a(n):= n*(a(n-1) + n) end: a(0):= 1:
seq(a(n), n=0..23);  # Alois P. Heinz, May 12 2021
  • Mathematica
    a=1;lst={a};Do[a=(a+n)*n;AppendTo[lst, a], {n, 2*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2008 *)
RecurrenceTable[{a[0]==1,a[n]==n(a[n-1]+n)},a,{n,20}] (* Harvey P. Dale, Feb 14 2015 *)

Formula

a(n) = A030297(n) + n! = A019460(2n+1) - n. - Henry Bottomley, Nov 10 2000
a(n)= n! + floor(2*e*n!-n) - 2, n>1. - Gary Detlefs, Jun 22 2010
E.g.f.: (1 + x * (1 + x) * exp(x)) / (1 - x). - Ilya Gutkovskiy, Oct 11 2021

Extensions

More terms from James Sellers, Apr 08 2000

A087297 Add the next prime and multiply by the next prime.

Original entry on oeis.org

1, 6, 27, 160, 1169, 12980, 168909, 2871742, 54563459, 1254960086, 36393843335, 1128209144346, 41743738342171, 1711493272030692, 73594210697321605, 3458927902774117644, 183323178847028237941, 10816067551974666042000
Offset: 0

Views

Author

Sebastian Gutierrez (sgutierr(AT)alum.mit.edu), Oct 26 2003

Keywords

Crossrefs

Cf. A019460.

Formula

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

Extensions

More terms from Ray Chandler, Oct 26 2003
Showing 1-10 of 11 results. Next

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

Content is available under The OEIS End-User License Agreement.