A064869 -id:A064869 - cp's OEIS frontend

A067151 Number of regions in regular n-gon which are quadrilaterals (4-gons) when all its diagonals are drawn.

0, 0, 6, 7, 24, 36, 90, 132, 168, 234, 378, 600, 672, 901, 954, 1444, 1580, 2520, 2860, 2990, 3696, 4800, 5070, 6750, 7644, 9309, 7920, 12927, 12896, 15576, 16898, 20475, 18684, 25382, 27246, 30966, 32760, 37064, 37170, 45838, 47300, 55350, 60996, 69231, 66864, 80507, 87550, 98124, 103272

Offset: 4

Sascha Kurz, Jan 06 2002

nonn

			a(6)=6 because the 6 regions around the center are quadrilaterals.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 4..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067163, A064869, A067152, A067153, A067154, A067155, A067156, A067157, A067158, A067159.

Conjecture: a(n) ~ c * n^4. Is c = 1/64 ? - Bill McEachen, Mar 03 2024

Title clarified, a(47) and above by Scott R. Shannon, Dec 04 2021

A064867 The minimal number which has multiplicative persistence 3 in base n.

Original entry on oeis.org

26, 63, 68, 23, 27, 31, 35, 39, 43, 46, 50, 54, 58, 62, 66, 69, 73, 77, 81, 85, 89, 92, 96, 100, 104, 108, 112, 115, 119, 123, 127, 131, 135, 138, 142, 146, 150, 154, 158, 161, 165, 169, 173, 177, 181, 184, 188, 192

Offset: 3

Sascha Kurz, Oct 08 2001

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.

			a(3) = 26 because 26 = [222]->[22]->[11]->[1] and no fewer n has persistence 3 in base 3.

Michael De Vlieger, Table of n, a(n) for n = 3..10000
M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
Sascha Kurz, Persistence in different bases
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
C. Rivera, Minimal prime with persistence p
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A003001, A031346, A064868, A064869, A064870, A064871, A064872.

With[{m = 3}, Table[Block[{k = 1}, While[Length@ FixedPointList[Times @@ IntegerDigits[#, n] &, k, 100] != m + 2, k++]; k], {n, 3, 5}]]~Join~Array[4 # - Floor[#/6] &, 45, 6] (* Michael De Vlieger, Aug 30 2021 *)

a(n) = 4*n-floor(n/6) for n > 5.
From Chai Wah Wu, Mar 07 2025: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n > 12.
G.f.: x^3*(48*x^9 - x^8 - 33*x^7 - 22*x^6 + 4*x^5 + 4*x^4 - 45*x^3 + 5*x^2 + 37*x + 26)/(x^7 - x^6 - x + 1). (End)

A064868 The minimal number which has multiplicative persistence 4 in base n.

Original entry on oeis.org

2344, 172, 131, 174, 52, 77, 75, 83, 75, 81, 89, 95, 101, 104, 110, 133, 143, 127, 133, 119, 124, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 238, 243, 248, 253, 258, 263, 268, 273, 278, 283

Offset: 5

Sascha Kurz, Oct 09 2001

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit. a(3) and a(4) do not seem to exist.

			a(6) = 172 because 172 = [444]->[144]->[24]->[12]->[2] and no lesser n has persistence 4 in base 6.

Michael De Vlieger, Table of n, a(n) for n = 5..10000
M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
Sascha Kurz, Persistence in different bases
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
C. Rivera, Minimal prime with persistence p
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A003001, A031346, A064867, A064869, A064870, A064871, A064872.

With[{m = 4, r = 24}, Table[Block[{k = 1}, While[Length@ FixedPointList[Times @@ IntegerDigits[#, n] &, k] != m + 2, k++]; k], {n, m + 1, r}]~Join~Array[(m + 1) # - Floor[#/r] &, 34, r + 1]] (* Michael De Vlieger, Aug 30 2021 *)

pers(nn, b) = {ok = 0; p = 0; until (ok, d = digits(nn, b); if (#d == 1, ok = 1, p++); nn = prod(k=1, #d, d[k]); if (nn == 0, ok = 1);); return (p);}
a(n) = {i=0; while (pers(i, n) != 4, i++); return (i);} \\ Michel Marcus, Jun 30 2013

a(n) = 5*n-floor(n/24) for n > 23.
From Chai Wah Wu, Mar 07 2025: (Start)
a(n) = a(n-1) + a(n-24) - a(n-25) for n > 48.
G.f.: x^5*(18*x^43 - x^42 + 21*x^41 - 5*x^40 - 18*x^39 - x^38 + 2*x^37 - x^36 - x^35 - 3*x^34 - x^33 + 13*x^32 - 3*x^31 + 7*x^30 - 20*x^29 + 127*x^28 - 38*x^27 + 46*x^26 + 2177*x^25 - 2339*x^24 + 5*x^23 + 5*x^22 + 5*x^21 + 5*x^20 - 14*x^19 + 6*x^18 - 16*x^17 + 10*x^16 + 23*x^15 + 6*x^14 + 3*x^13 + 6*x^12 + 6*x^11 + 8*x^10 + 6*x^9 - 8*x^8 + 8*x^7 - 2*x^6 + 25*x^5 - 122*x^4 + 43*x^3 - 41*x^2 - 2172*x + 2344)/(x^25 - x^24 - x + 1). (End)

Example modified by Harvey P. Dale, Oct 19 2022

A067152 Number of pentagonal regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 0, 7, 0, 18, 10, 44, 0, 117, 98, 150, 128, 357, 72, 646, 580, 903, 814, 1564, 840, 2050, 2106, 2862, 2128, 3625, 1440, 5146, 4896, 6105, 5542, 8190, 7452, 10471, 10184, 14235, 13160, 16564, 11382, 21156, 20548, 24300, 23920, 30362, 26112, 35231, 32700, 40341, 38532, 51834, 42012, 58905

Offset: 5

Sascha Kurz, Jan 06 2002

nonn

			a(5) = 1 because only the center-region is a pentagon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 5..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067164, A064869, A067151, A067153, A067154, A067155, A067156, A067157, A067158, A067159.

a(49) and beyond from Scott R. Shannon, Dec 04 2021

Definition clarified by N. J. A. Sloane, Jun 09 2025

A067153 Number of hexagonal regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 9, 0, 22, 0, 39, 0, 105, 48, 136, 18, 190, 120, 462, 66, 644, 72, 875, 390, 1296, 952, 1595, 450, 1891, 1472, 3201, 2346, 3640, 2124, 4773, 2698, 5577, 4000, 7298, 3444, 7912, 6336, 10980, 6532, 10904, 7824, 14651, 12150, 16779, 13260, 20299, 13176, 21560, 18200, 26961, 21634, 29500

Offset: 6

Sascha Kurz, Jan 06 2002

nonn

			a(9)=9 because drawing the regular 9-gon with all its diagonals yields 9 hexagons.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 6..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067165, A064869, A067151, A067152, A067154, A067155, A067156, A067157, A067158, A067159.

a(54) and beyond from Scott R. Shannon, Dec 04 2021

Definition clarified by N. J. A. Sloane, Jun 09 2025

A067154 Number of heptagonal regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 15, 0, 17, 18, 57, 0, 21, 44, 115, 0, 150, 104, 81, 112, 116, 0, 155, 224, 429, 306, 560, 180, 555, 836, 663, 640, 1025, 378, 1419, 660, 1710, 1564, 1786, 1200, 2352, 1050, 2754, 2236, 2597, 2700, 3410, 2240, 3078, 3190, 4602, 1860, 5551, 4898, 6363, 5056, 8515, 4950

Offset: 7

Sascha Kurz, Jan 06 2002

nonn

			a(7)=1 because the center-region is a heptagon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 7..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067166, A064869, A067151, A067152, A067153, A067155, A067156, A067157, A067158, A067159.

a(62) and beyond from Scott R. Shannon, Dec 04 2021

Definition clarified by N. J. A. Sloane, Jun 09 2025

A067155 Number of octagonal regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 0, 0, 13, 0, 0, 0, 34, 0, 38, 20, 0, 44, 23, 0, 50, 26, 108, 28, 145, 0, 217, 0, 264, 102, 315, 72, 407, 190, 546, 200, 656, 42, 903, 528, 810, 598, 1175, 288, 1078, 550, 1479, 780, 1166, 486, 1705, 784, 2451, 1276, 3068, 960, 3172, 1860, 4347, 2432, 4225, 2376, 4958, 2992, 3519, 2380

Offset: 8

Sascha Kurz, Jan 06 2002

nonn

			a(13)=13 because drawing the regular 13-gon and all its diagonals yields 13 octagons.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 8..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067167, A064869, A067151, A067152, A067153, A067154, A067156, A067157, A067158, A067159.

a(65) and beyond from Scott R. Shannon, Dec 04 2021

Definition clarified by Hugo Pfoertner, Dec 04 2021

A067156 Number of regions in regular n-gon which are 9-gons.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 123, 0, 0, 88, 45, 0, 0, 0, 0, 0, 51, 0, 0, 0, 165, 0, 114, 0, 118, 120, 61, 124, 0, 192, 195, 66, 67, 272, 138, 0, 568, 360, 146, 222, 600, 0, 231, 156, 237, 800, 567, 410, 664, 84, 255, 344, 174

Offset: 9

Sascha Kurz, Jan 06 2002

nonn

			a(9)=1 because drawing the regular 9-gon with all its diagonals yields 1 9-gon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 9..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067168, A064869, A067151, A067152, A067153, A067154, A067155, A067157, A067158, A067159.

a(86) and beyond by Scott R. Shannon, Dec 04 2021

A067157 Number of regions in regular n-gon which are 10-gons.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 87, 0, 62, 0, 0, 0, 0, 0, 74, 0, 0, 0, 41, 0, 0, 44, 0, 0, 235, 48, 147, 100, 51, 0, 159, 54, 110, 56, 114, 58, 177, 0, 183, 62, 378, 256, 195, 0, 134, 136, 621, 210, 71, 144, 438, 222, 750, 76, 385, 78, 1185, 80, 648, 82, 830, 336, 935

Offset: 10

Sascha Kurz, Jan 06 2002

nonn

			a(29)=87 because drawing the regular 29-gon with all its diagonals yields 87 10-gons.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 10..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067169, A064869, A067151, A067152, A067153, A067154, A067155, A067156, A067158, A067159.

a(83) and beyond by Scott R. Shannon, Dec 04 2021

A067158 Number of regions in regular n-gon which are 11-gons.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 71, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, 0, 0, 102, 103, 0, 0, 0, 0, 108, 0

Offset: 11

Sascha Kurz, Jan 06 2002

nonn

			a(11)=1 because drawing the regular 11-gon with all its diagonals yields 1 11-gon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 11..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A064869, A067151, A067152, A067153, A067154, A067155, A067156, A067157, A067159.

a(110) and beyond by Scott R. Shannon, Dec 04 2021

cp's OEIS Frontend

A067151 Number of regions in regular n-gon which are quadrilaterals (4-gons) when all its diagonals are drawn.

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A064867 The minimal number which has multiplicative persistence 3 in base n.

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A064868 The minimal number which has multiplicative persistence 4 in base n.

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A067152 Number of pentagonal regions in regular n-gon with all diagonals drawn.

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A067153 Number of hexagonal regions in regular n-gon with all diagonals drawn.

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A067154 Number of heptagonal regions in regular n-gon with all diagonals drawn.

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A067155 Number of octagonal regions in regular n-gon with all diagonals drawn.

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A067156 Number of regions in regular n-gon which are 9-gons.

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