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

Offset: 1

Author

Andrew McFarland, Jun 03 2008

Keywords

• nonn
• easy

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)

Examples

			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.
		

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

Crossrefs

Cf. A000554, A007304, A007997, A013929, A022998.

Cf. A047967, A129194, A235451, A285508, A321773.

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

Essentially the same as A004526.

Programs

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

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 *)

PARI

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

Python

def A140106(n): return n-2>>1 if n>1 else 0 # Chai Wah Wu, Sep 18 2023

SageMath

def A140106(n): return 0 if (n==1) else (n-2)//2
[A140106(n) for n in range(1,81)] # G. C. Greubel, Feb 10 2023

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

Extensions

More terms from Joseph Myers, Sep 05 2009