A000570 -id:A000570

Search: a000570 -id:a000570

Displaying 1-3 of 3 results found.

page 1

A001639

A Fielder sequence. a(n) = a(n-1) + a(n-3) + a(n-4) + a(n-5), n >= 6.
(Formerly M3353 N1349)

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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

T. D. Noe, Table of n, a(n) for n = 1..1000

Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 1, 1).

FORMULA

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

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 *)

PROG

(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))

(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

CROSSREFS

Cf. A000570.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by Michael Somos, Feb 17 2002

STATUS

approved

A121244

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Table of n, a(n) for n=1..27.

Eric Weisstein's World of Mathematics, Score Sequence.

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).

EXAMPLE

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.

KEYWORD

nonn

AUTHOR

Tanya Khovanova, Aug 22 2006

STATUS

approved

A121272

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

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).

LINKS

Table of n, a(n) for n=1..17.

Eric Weisstein's World of Mathematics, Score Sequence.

FORMULA

a(n) = A000568(n) - A000570(n). - Michel Marcus, Nov 01 2019

EXAMPLE

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.

CROSSREFS

Cf. A000568, A000570.

KEYWORD

nonn

AUTHOR

Tanya Khovanova, Aug 23 2006

STATUS

approved

page 1

Search completed in 0.009 seconds