A140106 - OEIS

A140106

Number of noncongruent diagonals in a regular n-gon.

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

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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

G. C. Greubel, Table of n, a(n) for n = 1..5000

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

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

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

a(n) = floor(A129194(n-1)/A022998(n)), for n > 1. - 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-François Alcover, Nov 17 2015 *)

PROG

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

(Magma)

A140106:= func< n | n eq 1 select 0 else Floor((n-2)/2) >;

[A140106(n): n in [1..80]]; // G. C. Greubel, Feb 10 2023

(SageMath)

def A140106(n): return 0 if (n==1) else (n-2)//2