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A006533
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Join n equal points around circle in all ways, count regions.
(Formerly M1118)
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18
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1, 2, 4, 8, 16, 30, 57, 88, 163, 230, 386, 456, 794, 966, 1471, 1712, 2517, 2484, 4048, 4520, 6196, 6842, 9109, 9048, 12951, 14014, 17902, 19208, 24158, 21510, 31931, 33888, 41449, 43826, 52956, 52992, 66712, 70034, 82993, 86840, 102091, 97776, 124314, 129448, 149986, 155894, 179447, 179280
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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This sequence and A007678 are two equivalent ways of presenting the same sequence. - N. J. A. Sloane, Jan 23 2020
In contrast to A007678, which only counts the polygons, this sequence also counts the n segments of the circle bounded by the arc of the circle and the straight line, both joining two neighboring points on the circle. Therefore, a(n) = A007678(n) + n. - M. F. Hasler, Dec 12 2021
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REFERENCES
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Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
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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T. D. Noe, Table of n, a(n) for n = 1..1000
M. F. Hasler, Interactive illustration of A006561(n) & A006533(n); colored version for n=6 and for n=8.
Sascha Kurz, m-gons in regular n-gons
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Burkard Polster (Mathologer), The hardest "What comes next?" (Euler's pentagonal formula), Youtube video, Oct 17 2020.
B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, No. 1 (1998), pp. 135-156. (Author's copy.).
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
M. Rubinstein, Drawings for n=4,5,6,...
Sequences formed by drawing all diagonals in regular polygon
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FORMULA
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Poonen and Rubinstein give an explicit formula for a(n) (see Mma code).
a(n) = A007678(n) + n. - T. D. Noe, Dec 23 2006
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MATHEMATICA
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del[m_, n_]:=If[Mod[n, m]==0, 1, 0];
R[n_]:=(n^4-6n^3+23n^2-18n+24)/24 + del[2, n](-5n^3+42n^2-40n-48)/48 - del[4, n](3n/4) + del[6, n](-53n^2+310n)/12 + del[12, n](49n/2) + del[18, n]*32n + del[24, n]*19n - del[30, n]*36n - del[42, n]*50n - del[60, n]*190n - del[84, n]*78n - del[90, n]*48n - del[120, n]*78n - del[210, n]*48n;
Table[R[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 A006533(n): return (1176*d(n, 12)*n - 3744*d(n, 120)*n + 1536*d(n, 18)*n - d(n, 2)*(5*n**3 - 42*n**2 + 40*n + 48) - 2304*d(n, 210)*n + 912*d(n, 24)*n - 1728*d(n, 30)*n - 36*d(n, 4)*n - 2400*d(n, 42)*n - 4*d(n, 6)*n*(53*n - 310) - 9120*d(n, 60)*n - 3744*d(n, 84)*n - 2304*d(n, 90)*n + 2*n**4 - 12*n**3 + 46*n**2 - 36*n)//48 + 1 # Chai Wah Wu, Mar 08 2021
(PARI) apply( {A006533(n)=if(n%2, (((n-6)*n+23)*n-18)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 19, 28) +!(n%6)*(310 -53*n))/12 +!(n%12)*49/2 +!(n%18)*32 +!(n%24)*19 -!(n%30)*36 -!(n%42)*50 -!(n%60)*190 -!(n%84)*78 -!(n%90)*48 -!(n%120)*78 -!(n%210)*48)*n)}, [1..44]) \\ M. F. Hasler, Aug 06 2021
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CROSSREFS
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Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.
Sequence in context: A164256 A164240 A164214 * A164188 A164186 A164187
Adjacent sequences: A006530 A006531 A006532 * A006534 A006535 A006536
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)
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EXTENSIONS
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Added more terms from b-file. - N. J. A. Sloane, Jan 23 2020
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STATUS
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approved
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