A332419 -id:A332419 - cp's OEIS frontend

A004277 1 together with positive even numbers.

1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132

Offset: 0

N. J. A. Sloane

Also number of non-attacking bishops on n X n board. - Koksal Karakus (karakusk(AT)hotmail.com), May 27 2002
Engel expansion of e^(1/2) (see A006784 for definition) [when offset by 1]. - Henry Bottomley, Dec 18 2000
Numbers n such that a 2n-group (i.e., a group of order 2n) has subgroup C_2. - Lekraj Beedassy, Oct 14 2004
Image of 1/(1-2x) under the mapping g(x)->g(x/(1+x^2)). - Paul Barry, Jan 16 2005
Position of n in A113322: A113322(a(n-1)) = n for n>0. - Reinhard Zumkeller, Oct 26 2005
Incrementally largest terms in the continued fraction for e. - Nick Hobson, Jan 11 2007
Conjecturally, the differences of two consecutive primes (without repetition). - Juri-Stepan Gerasimov, Nov 09 2009
Equals (1, 2, 2, 2, ...) convolved with (1, 0, 2, 0, 2, 0, 2, ...). - Gary W. Adamson, Mar 03 2010
a(n) is the number of 0-dimensional elements (vertices) in an n-cross polytope. - Patrick J. McNab, Jul 06 2015
Numbers k such that in the symmetric representation of sigma(k) there is no pair bars as its ends (Cf. A237593). - Omar E. Pol, Sep 28 2018
Also, the coordination sequence of the L-lattice (see A332419). - Sean A. Irvine, Jul 29 2020

E. Friedman, Math. Magic
Eric Weisstein's World of Mathematics, Cross Polytope
Index entries for sequences related to Engel expansions
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A004275, A008486, A030978, A097134.

INVERT transformation yields A098182 without A098182(0). - R. J. Mathar, Sep 11 2008

a004277 n = 2 * n - 1 + signum (1 - n)
a004277_list = 1 : [2, 4 ..]  -- Reinhard Zumkeller, Dec 19 2013

[1] cat [2*n: n in [1..80]]; // Vincenzo Librandi, Jul 11 2015

Join[{1}, Table[2*n, {n, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
Select[Range@ 105, PowerMod[#, #, # + 1] == 1 &] (* Robert G. Wilson v, Sep 26 2016 *)

G.f.: (1+x^2)/(1-x)^2. - Paul Barry, Feb 28 2003
Inverse binomial transform of Cullen numbers A002064. a(n)=2n+0^n. - Paul Barry, Jun 12 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1)*(-1)^k*2^(n-2k). - Paul Barry, Jan 16 2005
Equals binomial transform of [1, 1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Jul 15 2008
E.g.f.: 1+x*sinh(x) (aerated sequence). - Paul Barry, Oct 11 2009
a(n) = 0^n + 2*n = A000007(n) + A005843(n). - Reinhard Zumkeller, Jan 11 2012

Corrected by Charles R Greathouse IV, Mar 18 2010

A333139 The number of regions inside a decagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

220, 4220, 25220, 84280, 217800, 456640, 873090, 1501520, 2436020, 3736540, 5523970, 7830800, 10879460, 14665340, 19398660, 25173960, 32203320, 40502280, 50458120, 61995140, 75517160

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Decagon regions for n = 1.
Scott R. Shannon, Decagon regions for n = 2.
Scott R. Shannon, Decagon regions for n = 3.
Scott R. Shannon, Decagon regions with random distance-based coloring for n = 1.
Scott R. Shannon, Decagon regions with random distance-based coloring for n = 2.
Scott R. Shannon, Decagon regions with random distance-based coloring for n = 3.
Wikipedia, Decagon.

Cf. A332417 (n-gons), A332418 (vertices), A332419 (edges), A007678, A092867, A331452, A331929.

a(6)-a(21) from Lars Blomberg, May 18 2020

A332417 Irregular table read by rows: Take a decagon with all diagonals drawn, as in A333139. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

120, 90, 10, 2040, 1580, 460, 140, 10860, 8570, 4170, 1380, 210, 20, 10, 34360, 30420, 14240, 4020, 1120, 100, 20, 85600, 76920, 38610, 13360, 2650, 550, 110, 176760, 166400, 82560, 24500, 5500, 760, 140, 20, 327550, 320520, 159860, 51610, 10960, 2250, 300, 30, 0, 10

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See the links in A333139 for images of the decagons.

			A decagon with no other points along its edges, n = 1, contains 120 triangles, 90 quadrilaterals, 10 pentagons and no other n-gons, so the first row is [120, 90, 10]. A decagon with 1 point dividing its edges, n = 2, contains 2040 triangles, 1580 quadrilaterals, 460 pentagons, 140 hexagons and no other n-gons, so the second row is [2040,1580,460,140].
Table begins:
120, 90, 10;
2040,1580,460,140;
10860,8570,4170,1380,210,20,10;
34360,30420,14240,4020,1120,100,20;
85600,76920,38610,13360,2650,550,110;
176760, 166400, 82560, 24500, 5500, 760, 140, 20;
327550, 320520, 159860, 51610, 10960, 2250, 300, 30, 0, 10;
565060, 549520, 277360, 86540, 18960, 3560, 480, 20, 20;
910920, 891290, 447790, 147300, 32180, 5640, 720, 130, 40, 10;
The row sums are A333139.

Lars Blomberg, Table of n, a(n) for n = 1..181 (the first 21 rows)

Cf. A333139 (regions), A332418 (vertices), A332419 (edges), A007678, A092867, A331452, A331929.

a(29) and beyond from Lars Blomberg, May 18 2020

A332418 The number of vertices on a decagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

171, 3581, 23651, 80191, 213041, 444251, 862711, 1481141, 2413721, 3701951, 5493891, 7765621, 10833601, 14589491, 19315751, 25064491, 32107771, 40337021, 50328771, 61790891, 75318371

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See the links in A333139 for images of the decagons.

Cf. A333139 (regions), A332417 (n-gons), A332419 (edges), A330846, A092866, A332599, A007569.

a(6)-a(21) from Lars Blomberg, May 18 2020

A335802 a(n) is the number of edges formed by n-secting the angles of a decagon.

Original entry on oeis.org

10, 20, 310, 80, 950, 810, 1970, 390, 3130, 2980, 4750, 3300, 6730, 6450, 9190, 4650, 11570, 11200, 14450, 12040, 17810, 17350, 21650, 12370, 25190, 24660, 29450, 25940, 34210, 33510, 39450, 29090, 43890, 43540, 49810, 45080, 55830, 54950, 62430, 49210, 67610

Offset: 1

Lars Blomberg, Jun 25 2020

nonn

Cf. A332419 (n-sected sides, not angles), A335800 (regions), A335801 (vertices), A335803 (ngons).

A367324 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of edges in the resulting planar graph.

Original entry on oeis.org

3, 21, 8, 132, 92, 20, 429, 596, 290, 42, 1272, 1936, 2215, 708, 91, 2826, 6020, 7405, 4020, 1575, 136, 5640, 11088, 21150, 15120, 10962, 2632, 288, 10461, 26260, 43490, 38544, 35812, 17728, 5148, 390, 17094, 42144, 88230, 83136, 96257, 60672, 33291, 7800, 715

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Nov 14 2023

See A367322, A367323 and the cross references for images of the n-gons.

			The table begins:
3, 21, 132, 429, 1272, 2826, 5640, 10461, 17094, 26847, 41046, 61041, 84051, ...
8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, 183340, 222940, ...
20, 290, 2215, 7405, 21150, 43490, 88230, 151135, 250825, 384360, 578840, ...
42, 708, 4020, 15120, 38544, 83136, 169686, 294678, 475500, 746340, 1140624, ...
91, 1575, 10962, 35812, 96257, 201054, 389991, 668458, 1096508, 1675835, ...
136, 2632, 17728, 60672, 163776, 341920, 673112, 1155144, 1892528, 2905088, ...
288, 5148, 33291, 108252, 283464, 591723, 1133928, 1941786, 3166605, 4837824, ...
390, 7800, 48870, 164470, 430840, 900890, 1735800, 2982660, 4849740, 7438490, ...
715, 12793, 79134, 255552, 660033, 1376870, 2619287, 4482654, 7284904, ...
756, 16512, 99348, 346140, 912960, 1894920, 3685056, 6313164, 10261200, ...
1508, 26806, 160641, 516932, 1322802, 2757339, 5221996, 8932664, 14483183, ...
1722, 35546, 210658, 696682, 1773828, 3718400, 7030464, 12067720, 19517596, ...
2835, 49995, 292590, 939720, 2388825, 4976130, 9394815, 16064970, 26003640, ...
3088, 63456, 370784, 1217664, 3081472, 6455872, 12162640, 20861328, 33700320, ...
4896, 85680, 493017, 1579436, 3995102, 8318525, 15667336, 26783636, ...
4320, 99036, 593784, 1958922, 4978872, 10395450, 19644408, ...
7923, 137693, 781470, 2499792, 6298633, 13109658, 24645983, ...
8360, 167160, 941940, 3068280, 7705420, 16112480, 30238400, ...
12180, 210378, 1180683, 3772692, 9476418, 19717089, ...
12782, 252296, 1400674, 4547884, 11375584, 23776236, ...
17963, 308591, 1716306, 5478232, 13725457, 28550084, ...
16344, 350448, 1981416, 6460080, 16185624, ...
25600, 437700, 2415825, 7704700, 19262750, ...
.
.
.

Cf. A367322 (vertices), A367323 (regions), A274586 (1st row), A331448 (2nd row), A329710 (3rd row), A330845 (4th row), A333112 (5th row), A333110 (6th row), A332429 (7th row), A332419 (8th row), A135565 (1st column).

T(n,k) = A367322(n,k) + A367323(n,k) - 1 (Euler).

cp's OEIS Frontend

A004277 1 together with positive even numbers.

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A333139 The number of regions inside a decagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A332417 Irregular table read by rows: Take a decagon with all diagonals drawn, as in A333139. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A332418 The number of vertices on a decagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A335802 a(n) is the number of edges formed by n-secting the angles of a decagon.

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A367324 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of edges in the resulting planar graph.

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