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A007569
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Number of nodes in regular n-gon with all diagonals drawn.
(Formerly M0724)
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59
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1, 2, 3, 5, 10, 19, 42, 57, 135, 171, 341, 313, 728, 771, 1380, 1393, 2397, 1855, 3895, 3861, 6006, 5963, 8878, 7321, 12675, 12507, 17577, 17277, 23780, 16831, 31496, 30945, 40953, 40291, 52395, 47017, 66082, 65019, 82290, 80921, 101311, 84883, 123453, 121485
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OFFSET
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1,2
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COMMENTS
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I.e., vertex count of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018
Also the circumference of the n-polygon diagonal intersection graph (since these graphs are Hamiltonian). - Eric W. Weisstein, Mar 08 2018
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
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FORMULA
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MATHEMATICA
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del[m_, n_]:=If[Mod[n, m]==0, 1, 0]; Int[n_]:=If[n<4, n, n + Binomial[n, 4] + del[2, n](-5n^3+45n^2-70n+24)/24 - del[4, n](3n/2) + del[6, n](-45n^2+262n)/6 + del[12, n]*42n + del[18, n]*60n + del[24, n]*35n - del[30, n]*38n - del[42, n]*82n - del[60, n]*330n - del[84, n]*144n - del[90, n]*96n - del[120, n]*144n - del[210, n]*96n]; Table[Int[n], {n, 1, 1000}] (* T. D. Noe, Dec 21 2006 *)
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PROG
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(Python)
def d(n, m): return not n % m
def A007569(n): return 2 if n == 2 else n*(42*d(n, 12) - 144*d(n, 120) + 60*d(n, 18) - 96*d(n, 210) + 35*d(n, 24)- 38*d(n, 30) - 82*d(n, 42) - 330*d(n, 60) - 144*d(n, 84) - 96*d(n, 90)) + (n**4 - 6*n**3 + 11*n**2 + 18*n -d(n, 2)*(5*n**3 - 45*n**2 + 70*n - 24) - 36*d(n, 4)*n - 4*d(n, 6)*n*(45*n - 262))//24 # Chai Wah Wu, Mar 08 2021
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CROSSREFS
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KEYWORD
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easy,nonn,nice
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AUTHOR
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STATUS
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approved
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