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%I #52 Sep 18 2023 15:39:16
%S 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,
%T 14,15,15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,
%U 26,26,27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36,37
%N Number of noncongruent diagonals in a regular n-gon.
%C 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
%C Number of roots of the n-th Bernoulli polynomial in the left half-plane. - _Michel Lagneau_, Nov 08 2012
%C From _Gus Wiseman_, Oct 17 2020: (Start)
%C 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):
%C 111 211 221 222 322 332 333 433 443 444 544 554
%C 311 411 331 422 441 442 533 552 553 644
%C 511 611 522 622 551 633 661 662
%C 711 811 722 822 733 833
%C 911 A11 922 A22
%C B11 C11
%C (End)
%H G. C. Greubel, <a href="/A140106/b140106.txt">Table of n, a(n) for n = 1..5000</a>
%H Washington Bomfim, <a href="http://oeis.org/wiki/File:Ff2.png">Double-star corresponding to the partition [3,7]</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F a(n) = floor((n-2)/2), for n > 1, otherwise 0. - _Washington Bomfim_, Feb 12 2011
%F G.f.: x^4/(1-x-x^2+x^3). - _Colin Barker_, Jan 31 2012
%F a(n) = floor(A129194(n-1)/A022998(n)), for n > 1. - _Paul Curtz_, Jul 23 2017
%F a(n) = A001399(n-3) - A001399(n-6). Compare to A007997(n) = A001399(n-3) + A001399(n-6). - _Gus Wiseman_, Oct 17 2020
%e 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.
%p with(numtheory): for n from 1 to 80 do:it:=0:
%p 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:
%t a[1]=0; a[n_?OddQ] := (n-3)/2; a[n_] := n/2-1; Array[a, 100] (* _Jean-François Alcover_, Nov 17 2015 *)
%o (PARI) a(n)=if(n>1,n\2-1,0) \\ _Charles R Greathouse IV_, Oct 16 2015
%o (Magma)
%o A140106:= func< n | n eq 1 select 0 else Floor((n-2)/2) >;
%o [A140106(n): n in [1..80]]; // _G. C. Greubel_, Feb 10 2023
%o (SageMath)
%o def A140106(n): return 0 if (n==1) else (n-2)//2
%o [A140106(n) for n in range(1,81)] # _G. C. Greubel_, Feb 10 2023
%o (Python)
%o def A140106(n): return n-2>>1 if n>1 else 0 # _Chai Wah Wu_, Sep 18 2023
%Y Cf. A000554, A007304, A007997, A013929, A022998.
%Y Cf. A047967, A129194, A235451, A285508, A321773.
%Y A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
%Y Essentially the same as A004526.
%K nonn,easy
%O 1,6
%A _Andrew McFarland_, Jun 03 2008
%E More terms from _Joseph Myers_, Sep 05 2009
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