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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.

User: John Erickson

John Erickson's wiki page.

John Erickson has authored 3 sequences.

A342448 Partial sums of A066194.

Original entry on oeis.org

1, 3, 7, 10, 18, 25, 30, 36, 52, 67, 80, 94, 103, 113, 125, 136, 168, 199, 228, 258, 283, 309, 337, 364, 381, 399, 419, 438, 462, 485, 506, 528, 592, 655, 716, 778, 835, 893, 953, 1012, 1061, 1111, 1163, 1214, 1270, 1325, 1378, 1432, 1465, 1499, 1535, 1570
Offset: 1

Views

Author

John Erickson, Mar 12 2021

Keywords

Comments

n^2/2 + n/2 <= a(n) <= (31/50)*n^2 + n/2. The lower and upper bounds are attained at n=2^k and n=5*2^k for k >= 0.

Links

Crossrefs

Cf. A066194, A268836, A007814, A000120, A010060, A000035.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<2, n,
      Bits[Xor](n, b(iquo(n, 2))))
    end:
a:= proc(n) a(n):= 1+`if`(n<2, 0, a(n-1)+b(n-1)) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 14 2021
  • Mathematica
    a[1]=1;
a[n_/;EvenQ[n]]:= a[n] = 4a[n/2] - n/2;
a[n_/;OddQ[n]]:= a[n] = 2a[(n - 1)/2]+2a[(n + 1)/2]-(n-1)/2 - ThueMorse[n];
(* Second program: *)
b[n_] := If[n==0, 0, BitXor@@Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}]];
A066194 = Table[b[n]+1, {n, 0, 60}];
A066194 // Accumulate (* Jean-François Alcover, Sep 10 2022 *)

Formula

a(n) = A268836(n)/2 + n. - Kevin Ryde, Mar 12 2021
a(1) = 1; a(n) = [n == 0 (mod 2)]*(4*a(n/2) - n/2) + [n == 1 (mod 2)]*(2*a((n - 1)/2)+2*a((n + 1)/2)-(n-1)/2 - A010060(n)) where [] is an Iverson bracket

A337015 Number of distinct transitive subgroups of S_n, counting conjugates as distinct.

Original entry on oeis.org

1, 1, 2, 9, 20, 279, 512, 19087, 71602, 636365, 1517042, 321965982, 240609602, 8809543877, 144729615032, 26818608209252, 6603755558402, 2737593592637477
Offset: 1

Views

Author

John Erickson and Alexander Hulpke, Nov 21 2020

Keywords

Comments

This sequence is the labeled version of A002106. I have proven that A005432(p)-a(p) == 1 (mod p) if p is prime. Based on n<= 18,
I have conjectured that log(A005432(n)/a(n)) > (n-1)/2 for n prime and log(A005432(n)/a(n)) < (n-1)/2 for n composite.
L. Pyber shows c^{n^2*(1+o(1))} <= a(n) <= d^{n^2*(1+o(1))}, c=2^{1/16}, d=24^{1/6}; conjectures lower bound is accurate.

Examples

			For n = 4 the following 9 subgroups of S_4 are transitive:
Group( [ (1,4)(2,3), (1,3)(2,4) ] )
Group( [ (1,3,2,4), (1,2)(3,4) ] )
Group( [ (1,4,3,2), (1,3)(2,4) ] )
Group( [ (1,2,4,3), (1,4)(2,3) ] )
Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )
Group( [ (1,2)(3,4), (1,3)(2,4), (1,4) ] )
Group( [ (1,2)(3,4), (1,4)(2,3), (2,4) ] )
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )

Links

Crossrefs

Cf. A005432, A002106.

Programs

  • GAP
    NrTransSubSn:=function(n)
local s,cnt,i,u,no;
  s:=SymmetricGroup(n);
  cnt:=0;
  for i in [1..NrTransitiveGroups(n)] do
    u:=TransitiveGroup(n,i);
    no:=Normalizer(s,u);
    cnt:=cnt+IndexNC(s,no);
    Print("Class ",i,", found ",IndexNC(s,no)," new, total: ",cnt,"\n");
  od;
  return cnt;
end; # Alexander Hulpke

A320667 First differences of A066194.

Original entry on oeis.org

1, 2, -1, 5, -1, -2, 1, 10, -1, -2, 1, -5, 1, 2, -1, 21, -1, -2, 1, -5, 1, 2, -1, -10, 1, 2, -1, 5, -1, -2, 1, 42, -1, -2, 1, -5, 1, 2, -1, -10, 1, 2, -1, 5, -1, -2, 1, -21, 1, 2, -1, 5, -1, -2, 1, 10, -1, -2, 1, -5, 1, 2, -1, 85, -1, -2, 1, -5, 1, 2, -1, -10
Offset: 1

Views

Author

John Erickson, Oct 18 2018

Keywords

Examples

			To obtain the first 2^n-1 entries if you have the first 2^(n-1)-1 entries, adjoin 1/6 (-3 + (-1)^(1 + n) + 2^(2 + n)) to the right end of the list, negate the signs of the first 2^(n-1)-1 entries, and then adjoin that list to the right. For example for n=3 {1,2,-1} becomes {1,2,-1,5,-1,-2,1}.

Crossrefs

Cf. A066194.

Programs

  • Mathematica
    Fold[Join[#1, {#2}, -#1] &, {1},
Table[1/6 (-3 + (-1)^(1 + n) + 2^(2 + n)), {n, 2, 6}]]
t[n_/;IntegerQ[Log2[n]]]:=1/6 (-3 + (-1)^IntegerExponent[n,2] + 8*n);
t[n_/;Not[IntegerQ[Log2[n]]]]:=-t[n-2^Floor[Log2[n]]];
Table[t[j],{j,1,15}](* recursive formulation *)
Table[1/6 (-3+(-1)^IntegerExponent[j,2]+2^(IntegerExponent[j,2]+3))(-1)^(Total[IntegerDigits[j,2]]+1),{j,1,15}] (* closed form *)

Formula

a(n) = A066194(n+1) - A066194(n).
a(n) = (1/6)*(-3 + (-1)^A007814(n) + 2^(A007814(n)+3))*(-1)^(A000120(n)+1).

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