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.

A001639 A Fielder sequence. a(n) = a(n-1) + a(n-3) + a(n-4) + a(n-5), n >= 6.

Original entry on oeis.org

1, 1, 4, 9, 16, 22, 36, 65, 112, 186, 309, 522, 885, 1492, 2509, 4225, 7124, 12010, 20236, 34094, 57453, 96823, 163163, 274946, 463316, 780755, 1315687, 2217112, 3736129, 6295887, 10609441, 17878369, 30127497, 50768954, 85552651, 144167958, 242942778
Offset: 1

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Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000570.

Programs

  • Magma
    I:=[1, 1, 4, 9, 16]; [n le 5 select I[n] else Self(n-1) + Self(n-3) + Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 09 2018
  • Maple
    A001639:=-(1+3*z**2+4*z**3+5*z**4)/(-1+z+z**3+z**4+z**5); # conjectured by Simon Plouffe in his 1992 dissertation
  • Mathematica
    Drop[CoefficientList[Series[x*(1+3*x^2+4*x^3+5*x^4)/(1-x-x^3-x^4-x^5),{x,0,40}], x], 1] (* Stefan Steinerberger, Apr 10 2006 *)
LinearRecurrence[{1,0,1,1,1}, {1,1,4,9,16}, 30] (* G. C. Greubel, Jan 09 2018 *)
  • PARI
    a(n)=if(n<0,0,polcoeff(x*(1+3*x^2+4*x^3+5*x^4)/(1-x-x^3-x^4-x^5)+x*O(x^n),n))

Formula

G.f.: x*(1+3*x^2+4*x^3+5*x^4)/(1-x-x^3-x^4-x^5).

Extensions

Edited by Michael Somos, Feb 17 2002

A121244 Number of score vectors for tournaments on n nodes that do not determine the tournament uniquely.

Original entry on oeis.org

0, 0, 0, 0, 2, 11, 41, 136, 437, 1397, 4490, 14554, 47683, 158093, 530265, 1797631, 6153650, 21252343, 73986392, 259434758, 915667537, 3251026851, 11605063370, 41631062856, 150021553132, 542875085143, 1972049524959
Offset: 1

Views

Author

Tanya Khovanova, Aug 22 2006

Keywords

Examples

			For n = 3 there are two possible score sequences: {0,1,2} and {1,1,1}. Both of them uniquely define the corresponding tournament. Hence a(3) = 0.
The first occurrence of a sequence that doesn't define a tournament happens for n = 5. There are two such sequences {1,1,2,3,3} and {1,2,2,2,3}. Let's consider the first sequence: {1,1,2,3,3}. Let's take the two best players - the persons with 3 wins - as one of them should win the game with another, there is only one other person who won a game with one of the two best players. It could happen that this player has score 1 or 2. Thus we can get two different tournaments with the same score vector.

Links

Formula

This sequence is the difference between A000571 (Number of different scores that are possible in an n-team round-robin tournament) and A000570 (Number of tournaments on n nodes determined by their score vectors).

A121272 Number of outcomes of unlabeled n-team round-robin tournaments that are not uniquely defined by their score vectors.

Original entry on oeis.org

0, 0, 0, 0, 5, 45, 438, 6849, 191483, 9732967, 903753099, 154108310917, 48542114686488, 28401423719121589, 31021002160355165644, 63530415842308265098260, 244912778438520759443242406
Offset: 1

Views

Author

Tanya Khovanova, Aug 23 2006

Keywords

Comments

This sequence is the difference between A000568 (Number of outcomes of unlabeled n-team round-robin tournaments) and A000570 (Number of tournaments on n nodes determined by their score vectors).

Examples

			All tournaments with 4 or fewer teams are uniquely defined by their score vectors. Hence a(1) = a(2) = a(3) = a(4) = 0.
For five-team tournaments only two score sequences do not define the tournament uniquely: {1,1,2,3,3} and {1,2,2,2,3}. The first sequence corresponds to two different tournaments and the second sequence to three different tournaments. Thus a(5) = 5.

Links

Crossrefs

Cf. A000568, A000570.

Formula

a(n) = A000568(n) - A000570(n). - Michel Marcus, Nov 01 2019
Showing 1-3 of 3 results.

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

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