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A140106 Number of noncongruent diagonals in a regular n-gon. 15
0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Number of double-stars (diameter 3 trees) with n nodes. For n >= 3, number of partitions of n-2 into two parts. - Washington Bomfim, Feb 12 2011

Number of roots of the n-th Bernoulli polynomial in the left half-plane. - Michel Lagneau, Nov 08 2012

From Gus Wiseman, Oct 17 2020: (Start)

Also the number of 3-part non-strict integer partitions of n - 1. The Heinz numbers of these partitions are given by A285508. The version for partitions of any length is A047967, with Heinz numbers A013929. The a(4) = 1 through a(15) = 6 partitions are (A = 10, B = 11, C = 12):

  111  211  221  222  322  332  333  433  443  444  544  554

            311  411  331  422  441  442  533  552  553  644

                      511  611  522  622  551  633  661  662

                                711  811  722  822  733  833

                                          911  A11  922  A22

                                                    B11  C11

(End)

LINKS

Table of n, a(n) for n=1..76.

Washington Bomfim, Double-star corresponding to the partition [3,7]

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

Index entries for sequences related to trees

FORMULA

For n > 1, a(n) = floor((n-2)/2), otherwise 0. - Washington Bomfim, Feb 12 2011

G.f.: x^4/(1-x-x^2+x^3). - Colin Barker, Jan 31 2012

For n > 1, a(n) = floor(A129194(n - 1)/A022998(n)). - Paul Curtz, Jul 23 2017

a(n) = A001399(n-3) - A001399(n-6). Compare to A007997(n) = A001399(n-3) + A001399(n-6). - Gus Wiseman, Oct 17 2020

EXAMPLE

The square (n=4) has two congruent diagonals; so a(4)=1. The regular pentagon also has congruent diagonals; so a(5)=1. Among all the diagonals in a regular hexagon, there are two noncongruent ones; hence a(6)=2, etc.

MAPLE

with(numtheory): for n from 1 to 80 do:it:=0:

y:=[fsolve(bernoulli(n, x) , x, complex)] : for m from 1 to nops(y) do : if Re(y[m])<0 then it:=it+1:else fi:od: printf(`%d, `, it):od:

MATHEMATICA

a[1]=0; a[n_?OddQ] := (n-3)/2; a[n_] := n/2-1; Array[a, 100] (* Jean-Fran├žois Alcover, Nov 17 2015 *)

PROG

(PARI) a(n)=if(n>1, n\2-1, 0) \\ Charles R Greathouse IV, Oct 16 2015

CROSSREFS

Essentially the same as A004526.

Cf. A000554, A022998, A129194.

A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.

Cf. A007304, A007997, A013929, A047967, A235451, A285508, A321773.

Sequence in context: A076938 A080513 A004526 * A123108 A008619 A110654

Adjacent sequences:  A140103 A140104 A140105 * A140107 A140108 A140109

KEYWORD

nonn,easy

AUTHOR

Andrew McFarland, Jun 03 2008

EXTENSIONS

More terms from Joseph Myers, Sep 05 2009

STATUS

approved

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Last modified June 18 14:54 EDT 2021. Contains 345119 sequences. (Running on oeis4.)