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 1-3 of 3 results.

A107755 Numbers k such that Sum_{j=1..k} Catalan(j) == 0 (mod 3).

Original entry on oeis.org

2, 8, 12, 26, 30, 36, 38, 80, 84, 90, 92, 108, 110, 116, 120, 242, 246, 252, 254, 270, 272, 278, 282, 324, 326, 332, 336, 350, 354, 360, 362, 728, 732, 738, 740, 756, 758, 764, 768, 810, 812, 818, 822, 836, 840, 846, 848, 972, 974, 980, 984, 998, 1002, 1008, 1010
Offset: 1

Views

Author

N. J. A. Sloane, Jun 11 2005

Keywords

Links

Crossrefs

Cf. A000108, A107756, A107757, A108784, A137821, A137822, A137823, A137824.

Programs

  • Maple
    A107755 := proc(n) option remember ; local a; if n = 1 then 2; else for a from A107755(n-1)+1 do if add(A000108(k),k=1..a) mod 3 = 0 then RETURN(a) ; fi ; od: fi ; end: # R. J. Mathar, Feb 25 2008
c:=n->binomial(2*n,n)/(n+1): s:=0: for n from 1 to 1500 do s:=s+c(n): a[n]:=s mod 3: od: A:=[seq(a[n],n=1..1500)]: p:=proc(n) if A[n]=0 then n else fi end: seq(p(n),n=1..1500); # Emeric Deutsch, Jun 12 2005
  • Mathematica
    s0 = s2 = {}; s = 0; Do[s = Mod[s + (2 n)!/n!/(n + 1)!, 3]; Switch[ Mod[s, 3], 0, AppendTo[s0, n], 2, AppendTo[s2, n]], {n, 1055}]; s0 (* Robert G. Wilson v, Jun 14 2005 *)
Flatten[Position[Accumulate[CatalanNumber[Range[1100]]],?(Divisible[ #,3]&)]] (* _Harvey P. Dale, Feb 07 2016 *)
  • PARI
    n=0; s=Mod(0,3); A107755=vector(100,i, if( bitand(i,i-1), while(n++ && s+=binomial(2*n,n)/(n+1),), s=Mod(0,3);n=2*n+2+(log(i+.5)\log(2)%2)*2 ); /*print1(n",");*/ n) \\ M. F. Hasler, Feb 25 2008
  • PARI
    A107755(n)=sum( i=1,n, A137822(i) )*2 /* allows computation of a(10^4) in one second */ \\ M. F. Hasler, Mar 16 2008

Formula

a(2^j) = 2*a(2^j-1) + 2 (resp. + 4) if j is even (resp. odd). - M. F. Hasler, Feb 25 2008
a(n) = 2*Sum_{i=1..n} A137822(i). - M. F. Hasler, Mar 16 2008
{n: A137993(n-1) = 0}. - R. J. Mathar, Jul 07 2009

Extensions

More terms from Emeric Deutsch, Jun 12 2005
Corrected & extended by M. F. Hasler and R. J. Mathar, Feb 25 2008

A107757 Numbers k such that Sum_{j=1..k} Catalan(j) == 2 (mod 3).

Original entry on oeis.org

3, 9, 11, 27, 29, 35, 39, 81, 83, 89, 93, 107, 111, 117, 119, 243, 245, 251, 255, 269, 273, 279, 281, 323, 327, 333, 335, 351, 353, 359, 363, 729, 731, 737, 741, 755, 759, 765, 767, 809, 813, 819, 821, 837, 839, 845, 849, 971, 975, 981, 983, 999, 1001, 1007, 1011
Offset: 1

Views

Author

N. J. A. Sloane, Jun 11 2005

Keywords

Links

Crossrefs

Cf. A000108, A107755, A107756.
Equals A074939 - 1.

Programs

  • Maple
    c:=n->binomial(2*n,n)/(n+1): s:=0: for n from 1 to 1500 do s:=s+c(n): a[n]:=s mod 3: od: A:=[seq(a[n],n=1..1500)]: p:=proc(n) if A[n]=2 then n else fi end: seq(p(n),n=1..1500); # Emeric Deutsch, Jun 12 2005
  • Mathematica
    s0 = s2 = {}; s = 0; Do[s = Mod[s + (2 n)!/n!/(n + 1)!, 3]; Switch[ Mod[s, 3], 0, AppendTo[s0, n], 2, AppendTo[s2, n]], {n, 1055}]; s2 (* Robert G. Wilson v, Jun 14 2005 *)

Extensions

More terms from Emeric Deutsch, Jun 12 2005

A108784 Difference between A107757 and A107755.

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1
Offset: 1

Views

Author

Robert G. Wilson v, Jun 14 2005

Keywords

Comments

Of the 255 terms less than 10^4, 128 are positive.

Links

Crossrefs

Cf. A107755, A107756, A107757.

Programs

  • Maple
    Maple code from R. J. Mathar, Feb 25 2008:
A000108 := proc(n) option remember ; binomial(2*n,n)/(n+1) ; end:
A107757 := proc(n) option remember ; local a; if n = 1 then 3; else for a from A107757(n-1)+1 do if add(A000108(k),k=1..a) mod 3 = 2 then RETURN(a) ; fi ; od: fi ; end:
A107755 := proc(n) option remember ; local a; if n = 1 then 2; else for a from A107755(n-1)+1 do if add(A000108(k),k=1..a) mod 3 = 0 then RETURN(a) ; fi ; od: fi ; end:
A108784 := proc(n) A107757(n)-A107755(n) ; end: seq(A108784(n),n=1..120) ;
  • Mathematica
    s0 = s2 = {}; s = 0; Do[s = Mod[s + (2 n)!/n!/(n + 1)!, 3]; Switch[ Mod[s, 3], 0, AppendTo[s0, n], 2, AppendTo[s2, n]], {n, 10^4}]; s2 - s0

Formula

It appears that a(n) = A076826(2n)-1. - T. D. Noe, Jun 14 2007
a(n) = A107757(n) - A107755(n).

Extensions

Corrected by T. D. Noe, Jun 14 2007
Showing 1-3 of 3 results.

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

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