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 1-7 of 7 results.

A214939 Number of squarefree words of length n in a 5-ary alphabet.

Original entry on oeis.org

5, 20, 80, 300, 1140, 4260, 15960, 59580, 222600, 830880, 3102120, 11578800, 43220940, 161324400, 602159940, 2247585300, 8389237320, 31313155560, 116877700500, 436250537520
Offset: 1

Views

Author

R. H. Hardin Jul 30 2012

Keywords

Comments

All terms are multiples of 5 by symmetry. Michael S. Branicky, May 20 2021

Examples

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

Crossrefs

Column 4 of A214943.

Programs

  • Python
    from itertools import product
def a(n):
  if n == 1: return 5
  squares = ["".join(u) + "".join(u)
    for r in range(1, n//2 + 1) for u in product("01234", repeat = r)]
  words = ("0"+"".join(w) for w in product("01234", repeat=n-1))
  return 5*sum(all(s not in w for s in squares) for w in words)
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, May 20 2021

A214944 Number of squarefree words of length 5 in an (n+1)-ary alphabet.

Original entry on oeis.org

0, 30, 264, 1140, 3480, 8610, 18480, 35784, 64080, 107910, 172920, 265980, 395304, 570570, 803040, 1105680, 1493280, 1982574, 2592360, 3343620, 4259640, 5366130, 6691344, 8266200, 10124400, 12302550, 14840280, 17780364, 21168840, 25055130
Offset: 1

Views

Author

R. H. Hardin, Jul 30 2012

Keywords

Comments

Row 5 of A214943.

Examples

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

Links

Formula

Empirical: a(n) = n^5 + n^4 - 2*n^3 - n^2 + n.
Conjectures from Colin Barker, Jul 22 2018: (Start)
G.f.: 6*x^2*(5 + 14*x + x^2) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A214945 Number of squarefree words of length 6 in an (n+1)-ary alphabet.

Original entry on oeis.org

0, 42, 696, 4260, 16680, 50190, 126672, 281736, 569520, 1068210, 1886280, 3169452, 5108376, 7947030, 11991840, 17621520, 25297632, 35575866, 49118040, 66704820, 89249160, 117810462, 153609456, 198043800, 252704400, 319392450, 400137192
Offset: 1

Views

Author

R. H. Hardin, Jul 30 2012

Keywords

Comments

Row 6 of A214943.

Examples

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

Links

Crossrefs

Cf. A214943.

Formula

Empirical: a(n) = n^6 + n^5 - 3*n^4 - 2*n^3 + 2*n^2 + n.
Conjectures from Colin Barker, Jul 22 2018: (Start)
G.f.: 6*x^2*(7 + 67*x + 45*x^2 + x^3) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A214946 Number of squarefree words of length 7 in an (n+1)-ary alphabet.

Original entry on oeis.org

0, 60, 1848, 15960, 80040, 292740, 868560, 2218608, 5062320, 10575180, 20577480, 37769160, 66015768, 110690580, 179077920, 280842720, 428571360, 638388828, 930657240, 1330760760, 1869981960, 2586474660, 3526338288, 4744798800
Offset: 1

Views

Author

R. H. Hardin, Jul 30 2012

Keywords

Comments

Row 7 of A214943.

Examples

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

Links

Crossrefs

Cf. A214943.

Formula

Empirical: a(n) = n^7 + n^6 - 4*n^5 - 3*n^4 + 5*n^3 + 2*n^2 - 2*n.
Conjectures from Colin Barker, Jul 22 2018: (Start)
G.f.: 12*x^2*(5 + 114*x + 238*x^2 + 62*x^3 + x^4) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)

A214940 Number of squarefree words of length n in a 6-ary alphabet.

Original entry on oeis.org

6, 30, 150, 720, 3480, 16680, 80040, 383520, 1838160, 8807400, 42202560, 202209720, 968880960, 4642304520, 22243228680, 106576361760, 510651000360
Offset: 1

Views

Author

R. H. Hardin Jul 30 2012

Keywords

Comments

Column 5 of A214943

Examples

			Some solutions for n=6
..4....4....4....1....5....5....4....5....2....1....5....4....2....1....4....5
..0....5....5....4....2....4....1....1....1....3....3....1....4....2....1....2
..5....3....0....5....3....2....4....4....0....1....4....0....3....0....2....1
..3....4....3....3....2....5....2....5....5....4....0....3....5....4....4....0
..5....1....4....5....1....3....3....3....2....5....3....4....0....2....5....5
..0....0....5....2....3....4....2....5....3....2....4....2....2....4....1....1

Programs

  • Python
    from itertools import product
def a(n):
    if n == 1: return 6
    squares = ["".join(u) + "".join(u)
        for r in range(1, n//2 + 1) for u in product("012345", repeat=r)]
    words = ("0"+"".join(w) for w in product("012345", repeat=n-1))
    return 6*sum(all(s not in w for s in squares) for w in words)
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Jun 30 2021

A214941 Number of squarefree words of length n in a 7-ary alphabet.

Original entry on oeis.org

7, 42, 252, 1470, 8610, 50190, 292740, 1706250, 9946020, 57970080, 337883700, 1969343880, 11478292770, 66900843240, 389929523550, 2272690998690
Offset: 1

Views

Author

R. H. Hardin Jul 30 2012

Keywords

Comments

Column 6 of A214943

Examples

			Some solutions for n=6
..0....4....4....4....5....2....4....4....2....3....6....0....0....6....2....5
..4....1....6....0....1....5....1....1....0....5....4....5....1....0....1....0
..2....5....2....6....6....2....0....5....6....0....6....4....5....5....6....5
..3....0....6....5....0....1....3....4....2....3....2....2....3....3....3....2
..6....4....0....6....4....5....1....5....5....2....3....4....1....2....2....5
..4....0....6....4....6....2....0....6....2....1....4....6....5....4....6....0

A214942 Number of squarefree words of length n in an 8-ary alphabet.

Original entry on oeis.org

8, 56, 392, 2688, 18480, 126672, 868560, 5953248, 40806528, 279692784, 1917062784, 13139773584, 90061652352, 617293135536, 4231000268208, 28999772127648
Offset: 1

Views

Author

R. H. Hardin Jul 30 2012

Keywords

Comments

Column 7 of A214943.

Examples

			Some solutions for n=6:
..0....0....3....2....5....2....5....5....0....5....6....7....4....4....5....2
..5....1....5....3....2....0....3....6....7....6....7....6....7....7....4....3
..3....2....4....1....3....7....1....5....1....3....1....3....2....3....5....7
..0....0....0....4....4....4....0....1....7....0....6....2....3....4....1....4
..1....7....2....6....5....0....6....4....0....2....0....1....2....5....4....7
..3....6....4....4....2....1....0....2....7....0....1....4....7....7....7....2
Showing 1-7 of 7 results.

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