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A140106
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Number of noncongruent diagonals in a regular n-gon.
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23
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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
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1,6
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COMMENTS
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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
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)
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LINKS
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FORMULA
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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PROG
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(Magma)
A140106:= func< n | n eq 1 select 0 else Floor((n-2)/2) >;
(SageMath)
def A140106(n): return 0 if (n==1) else (n-2)//2
(Python)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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