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Three-column table read by rows: row n gives [number of triangle-triangle, triangle-quadrilateral, quadrilateral-quadrilateral] contacts for a row of n adjacent congruent rectangles divided by drawing diagonals of all possible rectangles (cf. A331452).
+20
2
4, 0, 0, 14, 8, 0, 20, 48, 4, 60, 80, 28, 68, 224, 68, 148, 368, 124, 224, 616, 268, 336, 1008, 420, 384, 1672, 648, 712, 2208, 972, 972, 3120, 1464, 1300, 4304, 1996, 1496, 6040, 2788, 2044, 7936, 3580, 2612, 10224, 4672, 3540, 12656, 5980, 4224, 16104, 7676, 5484, 19648, 9500
COMMENTS
For a row of n adjacent rectangles the only polygons formed when dividing all possible rectangles along their diagonals are 3-gons (triangles) and 4-gons (quadrilaterals). Hence the only possible edge-sharing contacts are 3-gons with 3-gons, 3-gons with 4-gons, and 4-gons with 4-gons. This sequence lists the number of these three possible combinations for a row of n adjacent rectangles. Note that the edges along the outside of the n adjacent rectangles are not counted as they are only in one n-gon.
FORMULA
Sum of row t = A331757(t) - 2(t + 1).
EXAMPLE
a(1) = 4, a(2) = 0, a(3) = 0. A single rectangle divided along its diagonals consists of four 3-gons, four edges, and no 4-gons. Therefore there are only four 3-gon-to-3-gon contacts. See the link image for n = 1.
a(4) = 14, a(5) = 8, a(6) = 0. Two adjacent rectangles divided along all diagonals consists of fourteen 3-gons and two 4-gons. The two 4-gons are separated and thus share all their edges, eight in total, with 3-gons. There are fourteen pairs of 3-gon-to-3-gon contacts. See the link image for n = 2.
a(7) = 20, a(8) = 48, a(9) = 4. Three adjacent rectangles divided along all diagonals consists of thirty-two 3-gons and fourteen 4-gons. There are two groups of three adjacent 4-gons, so there are four 4-gons-to-4-gon contacts. These, along with the other 4-gons, share 48 edges with 3-gons. There are also twenty 3-gon-to-3-gon contacts. See the link image for n = 3.
.
The table begins:
4,0,0;
14,8,0;
20,48,4;
60,80,28;
68,224,68;
148,368,124;
224,616,268;
336,1008,420;
384,1672,648;
712,2208,972;
972,3120,1464;
1300,4304,1996;
1496,6040,2788;
2044,7936,3580;
2612,10224,4672;
3540,12656,5980;
4224,16104,7676;
5484,19648,9500;
6568,24216,11936;
7836,29616,14468;
See A306302 for a count of the regions and images for other values of n.
Number of regions in regular n-gon with all diagonals drawn.
(Formerly M3411)
+10
143
0, 0, 1, 4, 11, 24, 50, 80, 154, 220, 375, 444, 781, 952, 1456, 1696, 2500, 2466, 4029, 4500, 6175, 6820, 9086, 9024, 12926, 13988, 17875, 19180, 24129, 21480, 31900, 33856, 41416, 43792, 52921, 52956, 66675, 69996, 82954, 86800, 102050, 97734, 124271, 129404, 149941
COMMENTS
This sequence and A006533 are two equivalent ways of presenting the same sequence. Also the circuit rank of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018 This sequence only counts polygons, in contrast to A006533, which also counts the n segments of the circumscribed circle delimited by the edges of the regular n-gon. Therefore, a(n) = A006533(n) - n. See also A006561 which counts the number of intersection points, and A350000 which considers iterated "cutting along diagonals". - M. F. Hasler, Dec 13 2021 The Petrie polygon orthographic projection of a regular n-simplex is a regular (n+1)-gon with all diagonals drawn. Hence a(n+1) is the number of regions in the Petrie polygon of a regular n-simplex. - Mohammed Yaseen, Nov 05 2022
REFERENCES
Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
C. A. Pickover, The Mathematics of Oz, Problem 58 "The Beauty of Polygon Slicing", Cambridge University Press, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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.
FORMULA
For odd n > 3, a(n) = sumstep {i=5, n, 2, (i-2)*floor(n/2)+(i-4)*ceiling(n/2)+1} + x*(x+1)*(2*x+1)/6*n), where x = (n-5)/2. Simplifying the floor/ceiling components gives the PARI code below. - Jon Perry, Jul 08 2003 For odd n, a(n) = (24 - 42*n + 23*n^2 - 6*n^3 + n^4)/24. - Graeme McRae, Dec 24 2004 For odd n, binomial transform of [1, 10, 29, 36, 16, 0, 0, 0, ...] = [1, 11, 50, 154, ...]. - Gary W. Adamson, Aug 02 2011 See the Mma code in A006533 for the explicit Poonen-Rubenstein formula that holds for all n. - N. J. A. Sloane, Jan 23 2020
MATHEMATICA
del[m_, n_]:=If[Mod[n, m]==0, 1, 0]; R[n_]:=If[n<3, 0, (n^4-6n^3+23n^2-42n+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 *)
PROG
(PARI) /* Only for odd n > 3, not suitable for other values of n! */ { a(n)=local(nr, x, fn, cn, fn2); nr=0; fn=floor(n/2); cn=ceil(n/2); fn2=(fn-1)^2-1; nr=fn2*n+fn+(n-2)*fn+cn; x=(n-5)/2; if (x>0, nr+=x*(x+1)*(2*x+1)/6*n); nr; } \\ Jon Perry, Jul 08 2003 (PARI) apply( { A007678(n)=if(n%2, (((n-6)*n+23)*n-42)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 31, 40) +!(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 (Python)
def d(n, m): return not n % m
def A007678(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 - 84*n)//48 + 1 # Chai Wah Wu, Mar 08 2021
CROSSREFS
A187781 gives number of distinct regions.
a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j).
+10
96
1, 8, 31, 80, 179, 332, 585, 948, 1463, 2136, 3065, 4216, 5729, 7568, 9797, 12456, 15737, 19520, 24087, 29308, 35315, 42120, 50073, 58920, 69025, 80264, 92871, 106756, 122475, 139528, 158681, 179608, 202529, 227400, 254597, 283784, 315957, 350576, 387977
COMMENTS
Also (1/4) * number of ways to select 3 distinct points forming a triangle of unsigned area = 1/2 from a square of grid points with side length n. Diagonal of triangle A320541. - Hugo Pfoertner, Oct 22 2018 Theorem: a(n) = n^2 + Sum_{i=2..n} (n+1-i)*(2*n+2-i)*phi(i).
Proof: Since gcd(n,n) = 1 if and only if n = 1, Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) = n^2 + Sum_{i=1..n, j=1..n, gcd(i,j)=1, (i,j) <> (1,1)} (n+1-i)*(n+1-j)
= n^2 + Sum_{i=2..n, j=1..i, gcd(i,j)=1} (n+1-i)*(n+1-j) + Sum_{j=2..n, i=1..j, gcd(i,j)=1} (n+1-i)*(n+1-j) = n^2 + 2*Sum_{i=2..n, j=1..i, gcd(i,j)=1} (n+1-i)*(n+1-j), i.e., the diagonal is not double-counted.
This is equal to n^2 + 2*Sum_{i=2..n, j is a totative of i} (n+1-i)*(n+1-j). Since Sum_{j is a totative of i} 1 = phi(i) and for i > 1, Sum_{j is a totative of i} j = i*phi(i)/2, the conclusion follows.
(End)
LINKS
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.
FORMULA
a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j).
a(n) = n^2 + Sum_{i=2..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 15 2021
MAPLE
local a, b, r ;
r := 0 ;
for a from 1 to n do
for b from 1 to n do
if igcd(a, b) = 1 then
r := r+(n+1-a)*(n+1-b);
end if;
end do:
end do:
r ;
end proc:
MATHEMATICA
a[n_] := Sum[(n-i+1) (n-j+1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
PROG
(Python)
from math import gcd
def a115004(n):
r=0
for a in range(1, n + 1):
for b in range(1, n + 1):
if gcd(a, b)==1:
r+=(n + 1 - a)*(n + 1 - b)
return r
print([a115004(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 21 2017 (Python)
from sympy import totient
def A115004(n): return n**2 + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2, n+1)) # Chai Wah Wu, Aug 15 2021 (PARI) a(n) = n^2 + sum(i=2, n, (n+1-i)*(2*n+2-i)*eulerphi(i)); \\ Michel Marcus, May 08 2024
Number of nodes in regular n-gon with all diagonals drawn.
(Formerly M0724)
+10
66
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
COMMENTS
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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.
MATHEMATICA
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 *)
PROG
(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
Number of regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles (a(0)=0 by convention).
+10
51
0, 4, 16, 46, 104, 214, 380, 648, 1028, 1562, 2256, 3208, 4384, 5924, 7792, 10052, 12744, 16060, 19880, 24486, 29748, 35798, 42648, 50648, 59544, 69700, 80992, 93654, 107596, 123374, 140488, 159704, 180696, 203684, 228624, 255892, 285152, 317400, 352096, 389576
COMMENTS
Assuming that the rectangles have vertices at (k,0) and (k,1), k=0..n, the projective map (x,y) -> ((1-y)/(x+1),y/(x+1)) maps their partition to the partition of the right isosceles triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. - Max Alekseyev, Apr 10 2019
FORMULA
a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) + n^2 + 2*n. - N. J. A. Sloane, Apr 11 2020 a(n) = 2n(n+1) + Sum_{i=2..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021
MAPLE
z := proc(n)
local a, b, r ;
r := 0 ;
for a from 1 to n do
for b from 1 to n do
if igcd(a, b) = 1 then
r := r+(n+1-a)*(n+1-b);
end if;
end do:
end do:
r ;
end proc:
a := n-> z(n)+n^2+2*n;
[seq(a(n), n=1..50)];
MATHEMATICA
z[n_] := Sum[(n - i + 1)(n - j + 1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
a[0] = 0;
a[n_] := z[n] + n^2 + 2n;
PROG
(Python)
from sympy import totient
def A306302(n): return 2*n*(n+1) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2, n+1)) # Chai Wah Wu, Aug 16 2021
CROSSREFS
See A331755 for the number of vertices, A331757 for the number of edges.
Number of intersections of diagonals in the interior of a regular n-gon.
(Formerly M3833)
+10
45
0, 0, 0, 1, 5, 13, 35, 49, 126, 161, 330, 301, 715, 757, 1365, 1377, 2380, 1837, 3876, 3841, 5985, 5941, 8855, 7297, 12650, 12481, 17550, 17249, 23751, 16801, 31465, 30913, 40920, 40257, 52360, 46981, 66045, 64981, 82251, 80881, 101270, 84841, 123410, 121441
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence) 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.
FORMULA
Let delta(m,n) = 1 if m divides n, otherwise 0.
For n >= 3, a(n) = binomial(n,4) + (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2,n)/24
- (3*n/2)*delta(4,n) + (-45*n^2 + 262*n)*delta(6,n)/6 + 42*n*delta(12,n)
+ 60*n*delta(18,n) + 35*n*delta(24,n) - 38*n*delta(30,n)
- 82*n*delta(42,n) - 330*n*delta(60,n) - 144*n*delta(84,n)
- 96*n*delta(90,n) - 144*n*delta(120,n) - 96*n*delta(210,n). [Poonen and Rubinstein, Theorem 1] - N. J. A. Sloane, Aug 09 2017 For odd n, a(n) = binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24, see A053126. For even n, use this formula, but then subtract 2 for every 3-crossing, subtract 5 for every 4-crossing, subtract 9 for every 5-crossing, etc. The number to be subtracted for a d-crossing is (d-1)*(d-2)/2. - Graeme McRae, Dec 26 2004
MAPLE
delta:=(m, n) -> if (n mod m) = 0 then 1 else 0; fi;
f:=proc(n) global delta;
if n <= 2 then 0 else \
binomial(n, 4) \
+ (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2, n)/24 \
- (3*n/2)*delta(4, n) \
+ (-45*n^2 + 262*n)*delta(6, n)/6 \
+ 42*n*delta(12, n) \
+ 60*n*delta(18, n) \
+ 35*n*delta(24, n) \
- 38*n*delta(30, n) \
- 82*n*delta(42, n) \
- 330*n*delta(60, n) \
- 144*n*delta(84, n) \
- 96*n*delta(90, n) \
- 144*n*delta(120, n) \
- 96*n*delta(210, n); fi; end;
MATHEMATICA
del[m_, n_]:=If[Mod[n, m]==0, 1, 0]; Int[n_]:=If[n<4, 0, 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 *)
PROG
(PARI) apply( { A006561(n)=binomial(n, 4)+if(n%2==0, (n>2) + (-5*n^2+45*n-70)*n/24 + vecsum([t[2] | t<-[4, 6, 12, 18, 24, 30, 42, 60, 84, 90, 120, 210; -3/2, (262-45*n)/6, 42, 60, 35, -38, -82, -330, -144, -96, -144, -96], n%t[1]==0])*n)}, [1..44]) \\ M. F. Hasler, Aug 23 2017, edited Aug 06 2021 (Python)
def d(n, m): return not n % m
def A006561(n): return 0 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 - 6*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
CROSSREFS
See A290447 for an analogous problem on a line.
Consider n equally spaced points along a line and join every pair of points by a semicircle above the line; a(n) is the number of intersection points.
+10
35
0, 0, 0, 1, 5, 15, 35, 70, 124, 200, 300, 445, 627, 875, 1189, 1564, 2006, 2568, 3225, 4035, 4972, 6030, 7250, 8701, 10323, 12156, 14235, 16554, 19124, 22072, 25250, 28863, 32827, 37166, 41949, 47142, 52653, 58794, 65503, 72741, 80437
COMMENTS
Only intersection points above the line are counted.
a(n) <= binomial(n,4) ( A000332), since that is the number of pairs of intersecting semicircles. See A290461 for the differences. The first time a triple intersection occurs is for n=9. Two fourfold intersections occur for n=13. - Torsten Sillke, Jul 27 2017
If the line is the x-axis and the two semicircles are for (x_1,0),(x_2,0) and (x_3,0),(x_4,0) (with x_1 < x_2, x_3 < x_4, and x_1 < x_3) then they intersect if and only if x_1 < x_3 < x_2 < x_4, and the intersection point has coordinates (x,y) with x=(x_3*x_4 - x_1*x_2) / (x_3 + x_4 - x_1 - x_2) and y^2 = (x_3-x_1)*(x_4-x_1)*(x_2-x_3)*(x_4-x_2) / (x_3 + x_4 - x_1 - x_2)^2. This allows identification of distinct (and duplicate) intersection points using only rational arithmetic. - David Applegate, Aug 07 2017 Suppose x_i are integers in the range 0 <= x_i < n. Then (x,y) is an intersection point if and only if (n-1-x,y) is an intersection point. Suppose x_4 < n-1. If (x,y) is an intersection point, then (i+x,y) is an intersection point for i = 1,..,n-1-x_4. - Chai Wah Wu, Aug 09 2017
REFERENCES
Torsten Sillke, email to N. J. A. Sloane, Jul 27 2017 (giving values for a(1)-a(13)).
LINKS
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence) 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.
PROG
(PARI) A290447(n, U=[])={for(A=1, n-3, for(C=A+1, n-2, for(B=C+1, n-1, for(D=B+1, n, U=setunion(U, [[(C*D-A*B)/(C+D-A-B), (C-A)*(D-A)*(C-B)*(D-B)/(C+D-A-B)^2]]))))); #U} \\ M. F. Hasler, Aug 07 2017 (Python)
from itertools import combinations
from fractions import Fraction
p, p2 = set(), set()
for b, c, d in combinations(range(1, n), 3):
e = b + d - c
f1, f2, g = Fraction(b*d, e), Fraction(b*d*(c-b)*(d-c), e**2), (n-1)*e - 2*b*d
for i in range(n-d):
if 2*i*e < g:
p2.add((i+f1, f2))
elif 2*i*e == g:
p.add(f2)
else:
break
CROSSREFS
See A006561 for an analogous problem on a circle.
Number of line segments in regular n-gon with all diagonals drawn.
+10
34
0, 1, 3, 8, 20, 42, 91, 136, 288, 390, 715, 756, 1508, 1722, 2835, 3088, 4896, 4320, 7923, 8360, 12180, 12782, 17963, 16344, 25600, 26494, 35451, 36456, 47908, 38310, 63395, 64800, 82368, 84082, 105315, 99972, 132756, 135014, 165243, 167720
COMMENTS
A line segment (or edge) is considered to end at any vertex where two or more chords meet.
I.e., edge count of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018
LINKS
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.
MATHEMATICA
del[m_, n_] := Boole[Mod[n, m] == 0];
If[n < 4, n,
n + Binomial[n, 4] + del[2, n] (-5 n^3 + 45 n^2 - 70 n + 24)/24 -
del[4, n] (3 n/2) + del[6, n] (-45 n^2 + 262 n)/6 +
del[12, n]*42 n + del[18, n]*60 n + del[24, n]*35 n -
del[30, n]*38 n - del[42, n]*82 n - del[60, n]*330 n -
del[84, n]*144 n - del[90, n]*96 n - del[120, n]*144 n -
del[210, n]*96 n];
If[n < 3,
0, (n^4 - 6 n^3 + 23 n^2 - 42 n + 24)/24 +
del[2, n] (-5 n^3 + 42 n^2 - 40 n - 48)/48 - del[4, n] (3 n/4) +
del[6, n] (-53 n^2 + 310 n)/12 + del[12, n] (49 n/2) +
del[18, n]*32 n + del[24, n]*19 n - del[30, n]*36 n -
del[42, n]*50 n - del[60, n]*190 n - del[84, n]*78 n -
del[90, n]*48 n - del[120, n]*78 n - del[210, n]*48 n];
Number of polygons formed by connecting all the 4n points on the perimeter of an n X n square by straight lines; a(0) = 0 by convention.
+10
33
0, 4, 56, 340, 1120, 3264, 6264, 13968, 22904, 38748, 58256, 95656, 120960, 192636, 246824, 323560, 425408, 587964, 682296, 932996, 1061232, 1327524, 1634488, 2049704, 2227672, 2806036, 3275800, 3810088, 4307520, 5298768
COMMENTS
There are n+1 points on each side of the square, but that counts the four corners twice, so there are a total of 4n points on the perimeter. - N. J. A. Sloane, Jan 23 2020 a(n) is always divisible by 4, by symmetry. If n is odd, a(n) is divisible by 8.
For n > 0, the vertices of the bounding square generate diametrical bisectors that cross at the center. Thus each diagram has fourfold symmetry.
For n > 0, an orthogonal n X n grid is produced by corresponding horizontal and vertical points on opposite sides.
Terms {1, 3, 9} are not congruent to 0 (mod 8).
Number of edges: {0, 8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, ...}. See A331448. (End)
EXAMPLE
For n = 3, the perimeter of the square contains 12 points:
* * * *
* *
* *
* * * *
Connect each point to every other point with a straight line inside the square. Then count the polygons (or regions) that have formed. There are 340 polygons, so a(3) = 340.
For n = 1, the full picture is:
*-*
|X|
*-*
The lines form four triangular regions, so a(1) = 4.
For n = 0, the square can be regarded as consisting of a single point, producing no lines or polygons, and so a(0) = 0.
Number of edges formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).
+10
25
28, 92, 240, 508, 944, 1548, 2520, 3754, 5392, 7528, 10296, 13570, 17844, 22768, 28584, 35704, 44048, 53380, 64728, 77292, 91500, 107828, 126408, 146772, 170080, 195580, 223764, 255010, 289792, 326996, 369320, 414908, 463880, 517724, 575404, 637530, 706172
COMMENTS
See A331452 for other illustrations.
LINKS
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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