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 A140106 Number of noncongruent diagonals in a regular n-gon. 23
 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 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 [A140106(n) for n in range(1, 81)] # G. C. Greubel, Feb 10 2023 (Python) def A140106(n): return n-2>>1 if n>1 else 0 # Chai Wah Wu, Sep 18 2023 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. Sequence in context: A065033 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 August 13 22:54 EDT 2024. Contains 375146 sequences. (Running on oeis4.)