A332953 -id:A332953 - cp's OEIS frontend

A333025 Irregular table read by rows: Take an isosceles triangle with its equal length sides divided into n equal parts with all diagonals drawn, as in A332953. Then T(n,k) = number of k-sided polygons in that figure for k>=3.

1, 5, 14, 3, 1, 29, 19, 4, 50, 66, 9, 81, 164, 12, 134, 313, 37, 2, 219, 546, 60, 7, 359, 853, 112, 9, 556, 1294, 160, 16, 1, 779, 1940, 283, 43, 3, 1105, 2780, 360, 53, 6, 1540, 3750, 670, 91, 5, 1, 2087, 5064, 873, 132, 11, 2806, 6625, 1144, 164, 7, 3

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 05 2020

See the links in A332953 for images of the triangles.

			Table begins:
1;
5;
14, 3, 1;
29, 19, 4;
50, 66, 9;
81, 164, 12;
134, 313, 37, 2;
219, 546, 60, 7;
359, 853, 112, 9;
556, 1294, 160, 16, 1;
779, 1940, 283, 43, 3;
1105, 2780, 360, 53, 6;
1540, 3750, 670, 91, 5, 1;
2087, 5064, 873, 132, 11;
2806, 6625, 1144, 164, 7, 3;
The row sums are A332953.

Lars Blomberg, Table of n, a(n) for n = 1..613 (the first 70 rows)

Cf. A332953 (regions), A333026 (vertices), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

A333027 The number of edges formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

Original entry on oeis.org

3, 10, 33, 96, 235, 486, 933, 1600, 2561, 3884, 5907, 8310, 11793, 15890, 20863, 27002, 35229, 44117, 55820, 68312, 82931, 100368, 121711, 143685, 169750, 199509, 232366, 268169, 312132, 355839, 409902, 465503, 527080, 596443, 668961, 746443, 839830, 937967

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 05 2020

nonn

See the links in A332953 for images of the triangles.

Lars Blomberg, Table of n, a(n) for n = 1..70

Cf. A332953 (regions), A333025 (n-gons), A333026 (vertices), A007678, A092867, A331452, A331911, A332357, A332358.

a(16) and beyond from Lars Blomberg, May 26 2020

A356984 Number of regions in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

Original entry on oeis.org

1, 4, 13, 28, 49, 70, 109, 148, 181, 244, 301, 334, 433, 508, 565, 676, 769, 811, 973, 1069, 1165, 1324, 1453, 1534, 1729, 1876, 1957, 2182, 2353, 2446, 2701, 2884, 3013, 3268, 3454, 3538, 3889, 4108, 4261, 4519, 4801, 4960, 5293, 5536, 5668, 6076, 6349, 6502, 6913, 7204, 7405, 7798, 8113

Offset: 0

Scott R. Shannon, Sep 08 2022

nonn

See A357007 for further images.

Scott R. Shannon, Table of n, a(n) for n = 0..250
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 5. This is the first term that forms intersections with non-simple vertices.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 50.
Scott R. Shannon, Image for n = 100.
Scott R. Shannon, Image for n = 200.
Talmon Silver, Classification of the intersection points and the number of regions

Cf. A357007 (vertices), A357008 (edges), A092867, A092098, A332953, A343755.

a(n) = A357008(n) - A357007(n) + 1 by Euler's formula.
Conjecture: a(n) = 3*n^2 + 1 for equilateral triangles that only contain simple vertices when cut by n internal equilateral triangles. This is never the case if (n + 1) mod 3 = 0 for n > 3.
a(n) = 1 + 3*n + T2(n) + 2*T3(n) + 3*T4(n); a(n) = 1 + 3*n^2 - T3(n) - 3*T4(n), where T2 is the number of internal vertices meeting exactly two segments (these vertices are labeled in the A357007 links as "4 ngons"), T3 is the number of internal vertices meeting exactly three segments ("6 ngons"), and T4 is the number of internal vertices meeting exactly four segments ("8 ngons"). - Talmon Silver, Sep 23 2022

A333026 The number of vertices formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

Original entry on oeis.org

3, 6, 16, 45, 111, 230, 448, 769, 1229, 1858, 2860, 4007, 5737, 7724, 10115, 13074, 17172, 21454, 27288, 33332, 40413, 48944, 59594, 70213, 82983, 97608, 113672, 131032, 152986, 174088, 201090, 228295, 258467, 292726, 328080, 365633, 412291, 460834, 512016

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 05 2020

nonn

See the links in A332953 for images of the triangles.

Lars Blomberg, Table of n, a(n) for n = 1..70

Cf. A332953 (regions), A333025 (n-gons), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

a(16) and beyond from Lars Blomberg, May 26 2020

A333519 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

Original entry on oeis.org

0, 2, 13, 48, 141, 312, 652, 1160, 1978, 3106, 4775, 6826, 9803, 13328, 17904, 23536, 30652, 38640, 48945, 60300, 74248, 89892, 108768, 128990, 153826, 180206, 211483, 245000, 284375, 325140, 374450, 425312, 484168, 545938, 616981, 690132, 775077, 862220

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 26 2020

nonn

A semicircular polygon with 2n+2 points is created by placing n+2 equally spaced vertices along the semicircle's arc (including the two end vertices). Also place n+2 equally spaced vertices along the diameter (again including the same two end vertices). Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Lars Blomberg, Table of n, a(n) for n = 0..50
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 20.

Cf. A007678, A290865, A255011, A332953, A333282, A334458, A334459.

a(21) and beyond from Lars Blomberg, May 01 2020

A125641 Square of the (3,1)-entry of the 3 X 3 matrix M^n, where M = [1,0,0; 1,1,0; 1,i,1].

Original entry on oeis.org

1, 5, 18, 52, 125, 261, 490, 848, 1377, 2125, 3146, 4500, 6253, 8477, 11250, 14656, 18785, 23733, 29602, 36500, 44541, 53845, 64538, 76752, 90625, 106301, 123930, 143668, 165677, 190125, 217186, 247040, 279873, 315877, 355250, 398196, 444925

Offset: 1

Gary W. Adamson, Nov 28 2006

Conjecture [False!]: Draw the segments joining every lattice point on axis X with every lattice point on axis Y for 1 <= x <= n and 1 <= y <= n. The number of regions formed with these segments and axis X and Y is a(n). - César Eliud Lozada, Feb 14 2013

The above conjecture appears to be wrong. The number of regions formed by this construction is given in A332953, which differs from this sequence for n > 5. - Scott R. Shannon, Mar 04 2020

			a(5)=25 because M^5 = [1,0,0; 5,1,0; 5+10i, 5i, 1] and |5+10i|^2 = 125.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
César Eliud Lozada, Counting regions [Warning: Although the drawings here appear to be correct for n <= 5, the generalization to higher n fails - see Comment above and A332953. - _N. J. A. Sloane_, Mar 04 2020]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A092098, A332953.

List([1..40],n-> n^2*(n^2-2*n+5)/4); # Muniru A Asiru, Feb 22 2019

[n^2*(n^2-2*n+5)/4: n in [1..40]]; // G. C. Greubel, Feb 22 2019

b[1]:=1: b[2]:=2+I: b[3]:=3+3*I: for n from 4 to 45 do b[n]:=3*b[n-1]-3*b[n-2]+b[n-3] od: seq(abs(b[j])^2,j=1..45);
with(linalg): M[1]:=matrix(3,3,[1,0,0,1,1,0,1,I,1]): for n from 2 to 45 do M[n]:=multiply(M[1],M[n-1]) od: seq(abs(M[j][3,1])^2,j=1..45);
seq(sum((binomial(n,m))^2,m=1..2),n=1..37); # Zerinvary Lajos, Jun 19 2008
# alternative Maple program:
a:= n-> abs((<<1|0|0>, <1|1|0>, <1|I|1>>^n)[3,1])^2:
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 09 2020

Table[n^2(n^2-2n+5)/4,{n,40}] (* Vincenzo Librandi, Feb 14 2012 *)

vector(40, n, n^2*(n^2-2*n+5)/4) \\ G. C. Greubel, Feb 22 2019

[n^2*(n^2-2*n+5)/4 for n in (1..40)] # G. C. Greubel, Feb 22 2019

a(n) = |b(n)|^2, where b(n) = 3b(n-1) - 3b(n-2) + b(n-3) for n >= 4; b(1)=1, b(2)=2+i, b(3)=3+3i (the recurrence relation follows from the minimal polynomial t^3 - 3t^2 + 3t - 1 of the matrix M).
a(n) = n^2*(n^2 - 2*n + 5)/4. - T. D. Noe, Feb 09 2007
O.g.f.: x*(1 + 3*x^2 + 2*x^3)/(1-x)^5. - R. J. Mathar, Dec 05 2007
a(n) = binomial(n,2)^2 + n^2, n > 1. - Gary Detlefs, Nov 23 2011
E.g.f.: x*(4 +6*x +4*x^2 +x^3)*exp(x)/4. - G. C. Greubel, Feb 22 2019

Edited by Emeric Deutsch, Dec 27 2006

Definition revised by N. J. A. Sloane, Mar 05 2020

cp's OEIS Frontend

A333025 Irregular table read by rows: Take an isosceles triangle with its equal length sides divided into n equal parts with all diagonals drawn, as in A332953. Then T(n,k) = number of k-sided polygons in that figure for k>=3.

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A333027 The number of edges formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

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A356984 Number of regions in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

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A333026 The number of vertices formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

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A333519 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

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A125641 Square of the (3,1)-entry of the 3 X 3 matrix M^n, where M = [1,0,0; 1,1,0; 1,i,1].

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