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.

A240437 Number of non-palindromic n-tuples of 5 distinct elements.

Original entry on oeis.org

0, 20, 100, 600, 3000, 15500, 77500, 390000, 1950000, 9762500, 48812500, 244125000, 1220625000, 6103437500, 30517187500, 152587500000, 762937500000, 3814695312500, 19073476562500, 95367421875000, 476837109375000, 2384185742187500, 11920928710937500, 59604644531250000, 298023222656250000
Offset: 1

Views

Author

Mikk Heidemaa, Aug 17 2014

Keywords

Examples

			For n=3 a(3)=100 solutions are:
{0,0,1}, {0,0,2}, {0,0,3}, {0,0,4}, {0,1,1}, {0,1,2}, {0,1,3}, {0,1,4},
{0,2,1}, {0,2,2}, {0,2,3}, {0,2,4}, {0,3,1}, {0,3,2}, {0,3,3}, {0,3,4},
{0,4,1}, {0,4,2}, {0,4,3}, {0,4,4}, {1,0,0}, {1,0,2}, {1,0,3}, {1,0,4},
{1,1,0}, {1,1,2}, {1,1,3}, {1,1,4}, {1,2,0}, {1,2,2}, {1,2,3}, {1,2,4},
{1,3,0}, {1,3,2}, {1,3,3}, {1,3,4}, {1,4,0}, {1,4,2}, {1,4,3}, {1,4,4},
{2,0,0}, {2,0,1}, {2,0,3}, {2,0,4}, {2,1,0}, {2,1,1}, {2,1,3}, {2,1,4},
{2,2,0}, {2,2,1}, {2,2,3}, {2,2,4}, {2,3,0}, {2,3,1}, {2,3,3}, {2,3,4},
{2,4,0}, {2,4,1}, {2,4,3}, {2,4,4}, {3,0,0}, {3,0,1}, {3,0,2}, {3,0,4},
{3,1,0}, {3,1,1}, {3,1,2}, {3,1,4}, {3,2,0}, {3,2,1}, {3,2,2}, {3,2,4},
{3,3,0}, {3,3,1}, {3,3,2}, {3,3,4}, {3,4,0}, {3,4,1}, {3,4,2}, {3,4,4},
{4,0,0}, {4,0,1}, {4,0,2}, {4,0,3}, {4,1,0}, {4,1,1}, {4,1,2}, {4,1,3},
{4,2,0}, {4,2,1}, {4,2,2}, {4,2,3}, {4,3,0}, {4,3,1}, {4,3,2}, {4,3,3},
{4,4,0}, {4,4,1}, {4,4,2}, {4,4,3}.

Links

Crossrefs

Cf. A233411, A242026, A242278.

Programs

  • Maple
    gf := (20*x^2) / (1 - 5*x - 5*x^2 + 25*x^3): ser := series(gf, x, 26):
seq(coeff(ser,x,n), n=1..25); # Peter Luschny, May 13 2019
  • Mathematica
    Table[1/2 * 5^(n/2) * ((Sqrt[5]-1) * (-1)^n - Sqrt[5]-1) + 5^n, {n, 25}]
  • PARI
    concat([0], Vec( ( (20*x^2) / (1 - 5*x - 5*x^2 + 25*x^3) + O(x^30) ) ) ) \\ Joerg Arndt, Aug 18 2014

Formula

a(n) = 1/2 * 5^(n/2) * ((sqrt(5)-1) * (-1)^n - sqrt(5)-1) + 5^n.
a(n) = 5^n - 5^ceiling(n/2).
a(n) = A000351(n) - A056451(n).
G.f.: (20*x^2) / (1 - 5*x - 5*x^2 + 25*x^3). [corrected by Peter Luschny, May 13 2019]

A242026 Number of non-palindromic n-tuples of 4 distinct elements.

Original entry on oeis.org

0, 12, 48, 240, 960, 4032, 16128, 65280, 261120, 1047552, 4190208, 16773120, 67092480, 268419072, 1073676288, 4294901760, 17179607040, 68719214592, 274876858368, 1099510579200, 4398042316800, 17592181850112, 70368727400448, 281474959933440, 1125899839733760
Offset: 1

Views

Author

Mikk Heidemaa, Aug 12 2014

Keywords

Comments

Non-palindromic vs palindromic (DNA) sequences (e.g., {a,c,a,c} is a non-palindromic sequence but {a,c,c,a} is palindromic). Useful in bioinformatics.

Examples

			For n=2 the a(2)=12 solutions (non-palindromic 2-tuples over 4 distinct elements {a,c,g,t}) are: {a,c}, {a,g}, {a,t}, {c,a}, {c,g}, {c,f}, {g,a},{g,c}, {g,t}, {t,a}, {t,c}, {t,g}.

Links

Crossrefs

Cf. A000302, A056450, A233411, A242278, A240437.

Programs

  • Mathematica
    Table[2^(n-1) * (2^(n+1) + (-1)^n - 3), {n, 66}]
LinearRecurrence[{4,4,-16},{0,12,48},30] (* Harvey P. Dale, May 24 2023 *)
  • PARI
    a(n) = ((-1)^n - 3)*2^(n-1) + 4^n; \\ Michel Marcus, Aug 12 2014
  • PARI
    concat(0, Vec(12*x^2 / ((2*x-1)*(2*x+1)*(4*x-1)) + O(x^100))) \\ Colin Barker, Aug 12 2014

Formula

a(n) = 2^(n-1) * (2^(n+1) + (-1)^n - 3).
a(n) = 4^n - 4^ceiling(n/2) = A000302(n) - A056450(n).
From Colin Barker, Aug 12 2014: (Start)
a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3).
G.f.: 12*x^2 / ((2*x-1)*(2*x+1)*(4*x-1)). (End)

Extensions

Typos in formula fixed by Colin Barker, Aug 12 2014

A251861 Number of non-palindromic words (length n>0) over the alphabet of 26 letters.

Original entry on oeis.org

0, 650, 16900, 456300, 11863800, 308898200, 8031353200, 208826607600, 5429491797600, 141167083772000, 3670344178072000, 95428956352766400, 2481152865171926400, 64509974695265340800, 1677259342076898860800, 43608742899220046995200, 1133827315379721221875200, 29479510200008489360729600, 766467265200220723378969600, 19928148895209267985244544000
Offset: 1

Views

Author

Mikk Heidemaa, Dec 10 2014

Keywords

Comments

Example: the acronyms 'OEIS' and 'SIEO' are two distinct non-palindromic words of length 4 among all possible such 456300 words (over 26 letters of the Latin alphabet).

Examples

			For n=2, the a(2)=650 solutions are {ab,ac,...,az,...,yz}, but not, e.g., 'aa' or 'zz'.

Links

Crossrefs

Analogs for other numbers of elements: (1) A000004, (2) A233411, (3) A242278, (4) A242026, (5) A240437.
Cf. A056450.

Programs

  • Maple
    seq(26^n - 26^ceil(n/2), n = 1 .. 50); # Robert Israel, Dec 11 2014
  • Mathematica
    f[n_, b_] := b^n - b^Ceiling[n/2]; Array[ f[#, 26] &, 50] (* Robert G. Wilson v, Dec 10 2014 *)
Table[2^(n/2-1)*13^(n/2)*((-1)^n*(Sqrt[26]-1)-Sqrt[26]-1)+26^n, {n, 50}]
  • PARI
    a(n)=26^n-26^ceil(n/2) \\ Charles R Greathouse IV, Dec 10 2014

Formula

a(n) = 2^(n/2-1)*13^(n/2)*((-1)^n*(sqrt(26)-1)-sqrt(26)-1)+26^n.
a(n) = 26^n - 26^ceiling(n/2).
G.f.: 650*x^2/((1 - 26*x)*(1 - 26*x^2)).
a(n+3) = 26*a(n+2) + 26*a(n+1) - 676*a(n). - Robert Israel, Dec 11 2014
Showing 1-3 of 3 results.

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