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 1-4 of 4 results.

A184294 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..7 arrays.

Original entry on oeis.org

8, 36, 36, 176, 1072, 176, 1044, 43800, 43800, 1044, 6560, 2098720, 14913536, 2098720, 6560, 43800, 107377488, 5726645688, 5726645688, 107377488, 43800, 299600, 5726689312, 2345624810432, 17592189193216, 2345624810432, 5726689312, 299600
Offset: 1

Views

Author

R. H. Hardin, Jan 10 2011

Keywords

Examples

			Table starts
       8         36           176           1044          6560      43800
      36       1072         43800        2098720     107377488 5726689312
     176      43800      14913536     5726645688 2345624810432
    1044    2098720    5726645688 17592189193216
    6560  107377488 2345624810432
   43800 5726689312
  299600

Links

Crossrefs

Columns 1-3 are A054627, A184292, A184293.
Cf. A184271, A184284.

Programs

  • Maple
    with(numtheory):
T:= (n, k)-> add(add(phi(c)*phi(d)*8^(n*k/ilcm(c, d)),
             c=divisors(n)), d=divisors(k))/(n*k):
seq(seq(T(n, 1+d-n), n=1..d), d=1..8);  # Alois P. Heinz, Aug 20 2017
  • Mathematica
    T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*8^(n*(k/LCM[c, d])), {d, Divisors[k]}], {c, Divisors[n]}]; Table[T[n - k + 1, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 30 2017, after Alois P. Heinz *)
  • PARI
    T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 8^(n*k/lcm(c,d)))); \\ Andrew Howroyd, Sep 27 2017

Formula

T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 8^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

A054615 a(n) = Sum_{d|n} phi(d)*8^(n/d).

Original entry on oeis.org

0, 8, 72, 528, 4176, 32800, 262800, 2097200, 16781472, 134218800, 1073774880, 8589934672, 68719748256, 549755813984, 4398048608688, 35184372156480, 281474993496384, 2251799813685376, 18014398644225456, 144115188075856016
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Column k=8 of A185651.
Cf. A054627.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*8^(n/d)); \\ Michel Marcus, Jul 11 2021

Formula

a(n) = Sum_{k=1..n} 8^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

A121775 T(n, k) = Sum_{d|n} phi(n/d)*binomial(d,k) for n>0, T(0, 0) = 1. Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 5, 3, 1, 4, 8, 7, 4, 1, 5, 9, 10, 10, 5, 1, 6, 15, 20, 21, 15, 6, 1, 7, 13, 21, 35, 35, 21, 7, 1, 8, 20, 36, 60, 71, 56, 28, 8, 1, 9, 21, 42, 86, 126, 126, 84, 36, 9, 1, 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1, 11, 21, 55, 165, 330, 462, 462, 330, 165, 55
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2006

Keywords

Comments

For n>0, (1/n)*Sum_{k=0..n} T(n,k)*(c-1)^k is the number of n-bead necklaces with c colors. See the cross references.

Examples

			Triangle begins:
[ 0]  1;
[ 1]  1,  1;
[ 2]  2,  3,  1;
[ 3]  3,  5,  3,   1;
[ 4]  4,  8,  7,   4,   1;
[ 5]  5,  9, 10,  10,   5,   1;
[ 6]  6, 15, 20,  21,  15,   6,   1;
[ 7]  7, 13, 21,  35,  35,  21,   7,   1;
[ 8]  8, 20, 36,  60,  71,  56,  28,   8,  1;
[ 9]  9, 21, 42,  86, 126, 126,  84,  36,  9,  1;
[10] 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1;

Crossrefs

Cf. A053635 (row sums), A121776 (antidiagonal sums), A054630, A327029.
Cf. A000031 (c=2), A001867 (c=3), A001868 (c=4), A001869 (c=5), A054625 (c=6), A054626 (c=7), A054627 (c=8), A054628 (c=9), A054629 (c=10).

Programs

  • PARI
    T(n,k)=if(n
  • SageMath
    # uses[DivisorTriangle from A327029]
DivisorTriangle(euler_phi, binomial, 13) # Peter Luschny, Aug 24 2019

A161222 Consider necklaces with n beads, each of one of four colors (say C1, C2, C3, C4), where the n segments of cord between the beads are each colored red or green; a(n) is the number of different necklaces under the action of the dihedral group D_{2n}.

Original entry on oeis.org

1, 8, 30, 120, 618, 3536, 22668, 151848, 1054986, 7472984, 53737896, 390582648, 2863716060, 21145502960, 157076310324, 1172820793824, 8796118712586, 66229473393728, 500400163666188, 3792505486235544, 28823039252629512, 219604100410657136, 1676976747053723292
Offset: 0

Views

Author

H. O. Pollak (hpollak(AT)adsight.com) and N. J. A. Sloane, Nov 21 2009

Keywords

Comments

If the group is changed to C_n we get A054627.

Crossrefs

Cf. A161221, A054627.

Programs

  • Maple
    with(numtheory); f:=proc(n) local t1,d,m;
if n mod 2 = 0 then m:=n/2; t1:=3*2^(3*m);
else m:=(n-1)/2; t1:=2^(3*m+3); fi;
(1/2)*( (1/n) * add( phi(d)*2^(3*n/d), d in divisors(n)) + t1 );
end; # this assumes n>0

Formula

For formula see Maple code.
Showing 1-4 of 4 results.

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

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