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 1-5 of 5 results.

A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 3, 4, 4, 3, 2, 8, 6, 8, 2, 4, 4, 6, 6, 4, 4, 2, 10, 4, 15, 4, 10, 2, 4, 4, 12, 6, 6, 12, 4, 4, 3, 11, 4, 16, 8, 16, 4, 11, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 10, 10, 22, 4, 30, 4, 22, 10, 10, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6
Offset: 1

Views

Author

Peter Luschny, Sep 12 2012

Keywords

Comments

T(n,k) = number of subgroups of C_n X C_k. [Hampjes et al.] - N. J. A. Sloane, Feb 02 2013

Examples

			[----1---2---3---4---5---6---7---8---9--10--11--12]
[ 1] 1,  2,  2,  3,  2,  4,  2,  4,  3,  4,  2,  6
[ 2] 2,  5,  4,  8,  4, 10,  4, 11,  6, 10,  4, 16
[ 3] 2,  4,  6,  6,  4, 12,  4,  8, 10,  8,  4, 18
[ 4] 3,  8,  6, 15,  6, 16,  6, 22,  9, 16,  6, 30
[ 5] 2,  4,  4,  6,  8,  8,  4,  8,  6, 16,  4, 12
[ 6] 4, 10, 12, 16,  8, 30,  8, 22, 20, 20,  8, 48
[ 7] 2,  4,  4,  6,  4,  8, 10,  8,  6,  8,  4, 12
[ 8] 4, 11,  8, 22,  8, 22,  8, 37, 12, 22,  8, 44
[ 9] 3,  6, 10,  9,  6, 20,  6, 12, 23, 12,  6, 30
[10] 4, 10,  8, 16, 16, 20,  8, 22, 12, 40,  8, 32
[11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 14, 12
[12] 6, 16, 18, 30, 12, 48, 12, 44, 30, 32, 12, 90
.
Displayed as a triangular array:
1,
2,  2,
2,  5,  2,
3,  4,  4,  3,
2,  8,  6,  8, 2,
4,  4,  6,  6, 4,  4,
2, 10,  4, 15, 4, 10, 2,
4,  4, 12,  6, 6, 12, 4,  4,
3, 11,  4, 16, 8, 16, 4, 11, 3,

Links

Crossrefs

Cf. A216620, A216621, A216622, A216623, A216625, A216626, A216627.
Main diagonal is A060724.
Rows give A000005, A060710, A054584, A221951, A221852.

Programs

  • Maple
    with(numtheory):
T:= (n, k)-> add(add(igcd(c,d), c=divisors(n)), d=divisors(k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..14); # Alois P. Heinz, Sep 12 2012
T:=proc(m,n) local d; add( d*tau(m*n/d^2), d in divisors(gcd(m,n))); end; # N. J. A. Sloane, Feb 02 2013
  • Mathematica
    t[n_, k_] := Sum[Sum[GCD[c, d], {c, Divisors[n]}], {d, Divisors[k]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 21 2013 *)
  • Sage
    def A216624(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(gcd, cp))
for n in (1..12): [A216624(n,k) for k in (1..12)]

Formula

T(n,n) = A060724(n) = sum_{d|n} d*tau((n/d)^2).
T(n,1) = T(1,n) = A000005(n) = tau(n).
T(n,2) = T(2,n) = A060710(n) = sum_{d|n} (3-[d is odd]) (Iverson bracket).
T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
T(prime(n),1) = A007395(n) = 2.
T(prime(n),prime(n)) = A113935(n) = prime(n)+3.

A221850 Number of nX3 arrays of occupancy after each element stays put or moves to some horizontal, diagonal or antidiagonal neighbor, without consecutive moves in the same direction.

Original entry on oeis.org

6, 302, 10085, 354670, 12277568, 426752551, 14818941152, 514719294830, 17877010521736, 620906972675376, 21565339027443844, 749008098922831183, 26014567709115430632, 903538668696429777028, 31381728865914983789375
Offset: 1

Views

Author

R. H. Hardin Jan 27 2013

Keywords

Comments

Column 3 of A221852

Examples

			Some solutions for n=3
..1..0..1....0..2..0....1..1..0....1..1..2....0..1..2....0..1..1....1..0..1
..0..0..1....1..0..0....1..6..0....2..0..0....0..3..0....1..0..1....0..0..1
..1..2..3....3..2..1....0..0..0....0..1..2....2..0..1....2..1..2....0..4..2

Links

A221853 Number of 2 X n arrays of occupancy after each element stays put or moves to some horizontal, diagonal or antidiagonal neighbor, without consecutive moves in the same direction.

Original entry on oeis.org

1, 31, 302, 3437, 37155, 406612, 4434129, 48397883, 528123314, 5763315281, 62892896491, 686329748776, 7489683895265, 81732408497079, 891918279715066, 9733204380907645, 106215187016111667, 1159090627331995148
Offset: 1

Views

Author

R. H. Hardin, Jan 27 2013

Keywords

Comments

Row 2 of A221852.

Examples

			Some solutions for n=3:
..3..2..1....2..0..2....0..2..1....0..0..1....2..0..0....2..1..1....0..2..0
..0..0..0....1..0..1....0..3..0....1..2..2....1..3..0....0..0..2....3..0..1

Links

Crossrefs

Cf. A221852.

Formula

Empirical: a(n) = 13*a(n-1) - 14*a(n-2) - 113*a(n-3) + 189*a(n-4) - 10*a(n-5) - 42*a(n-6) + 18*a(n-7) - 2*a(n-8).
Empirical g.f.: x*(1 + 18*x - 87*x^2 + 58*x^3 + 16*x^4 - 8*x^5 - 2*x^6) / (1 - 13*x + 14*x^2 + 113*x^3 - 189*x^4 + 10*x^5 + 42*x^6 - 18*x^7 + 2*x^8). - Colin Barker, Aug 11 2018

A221851 Number of nX4 arrays of occupancy after each element stays put or moves to some horizontal, diagonal or antidiagonal neighbor, without consecutive moves in the same direction.

Original entry on oeis.org

13, 3437, 465305, 67228149, 9607602488, 1374453486067, 196583533871825, 28117377558933453
Offset: 1

Views

Author

R. H. Hardin Jan 27 2013

Keywords

Comments

Column 4 of A221852

Examples

			Some solutions for n=3
..0..2..1..0....0..2..0..0....0..1..1..0....0..3..0..2....0..4..0..2
..0..0..3..0....0..1..7..0....1..0..1..4....0..1..0..2....0..0..1..0
..1..3..2..0....1..0..1..0....0..3..0..1....0..1..1..2....0..3..1..1

A221854 Number of 3Xn arrays of occupancy after each element stays put or moves to some horizontal, diagonal or antidiagonal neighbor, without consecutive moves in the same direction.

Original entry on oeis.org

1, 306, 10085, 465305, 19159028, 811781250, 34109464173, 1436352539452, 60446151885851, 2544184795092968
Offset: 1

Views

Author

R. H. Hardin Jan 27 2013

Keywords

Comments

Row 3 of A221852

Examples

			Some solutions for n=3
..2..1..1....0..1..0....0..0..1....2..1..0....1..1..3....1..0..0....1..0..2
..0..0..0....0..3..2....0..4..1....1..0..1....0..0..1....1..3..1....3..0..1
..3..1..1....0..3..0....0..1..2....1..3..0....0..2..1....2..1..0....0..2..0
Showing 1-5 of 5 results.

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

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