A001859 -id:A001859 - cp's OEIS frontend

A184010 n + floor(sqrt(-1+4n/3)); complement of A001859 (except for initial zero).

1, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111

Offset: 1

Clark Kimberling, Jan 08 2011

Cf. A001859.

  a=4/3; b=-1; Table[n+Floor[(a*n+b)^(1/2)],{n,100}]

a(n) = n + floor(sqrt(-1+4n/3))

A045944 Rhombic matchstick numbers: a(n) = n(3n+2).

Original entry on oeis.org

0, 5, 16, 33, 56, 85, 120, 161, 208, 261, 320, 385, 456, 533, 616, 705, 800, 901, 1008, 1121, 1240, 1365, 1496, 1633, 1776, 1925, 2080, 2241, 2408, 2581, 2760, 2945, 3136, 3333, 3536, 3745, 3960, 4181, 4408, 4641, 4880, 5125, 5376, 5633, 5896, 6165, 6440

Offset: 0

R. K. Guy

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the n-th term of the sequence found by reading the line from 0 in the direction 0,5,.... The spiral begins:
                 .
                 85--84--83--82--81--80
                   .                   \
                   56--55--54--53--52  79
                   / .               \   \
                 57  33--32--31--30  51  78
                 /   / .           \   \   \
               58  34  16--15--14  29  50  77
               /   /   / .       \   \   \   \
             59  35  17   5---4  13  28  49  76
             /   /   /   / .   \   \   \   \   \
           60  36  18   6   0   3  12  27  48  75
           /   /   /   /   /   /   /   /   /   /
         61  37  19   7   1---2  11  26  47  74
           \   \   \   \         /   /   /   /
           62  38  20   8---9--10  25  46  73
             \   \   \             /   /   /
             63  39  21--22--23--24  45  72
               \   \                 /   /
               64  40--41--42--43--44  71
                 \                     /
                 65--66--67--68--69--70
(End)
Connection to triangular numbers: a(n) = 4*T_n + S_n where T_n is the n-th triangular number and S_n is the n-th square. - William A. Tedeschi, Sep 12 2010
Also, second octagonal numbers. - Bruno Berselli, Jan 13 2011
Sequence found by reading the line from 0, in the direction 0, 16, ... and the line from 5, in the direction 5, 33, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012
Let P denote the points from the n X n grid. A(n-1) also coincides with the minimum number of points Q needed to "block" P, that is, every line segment spanned by two points from P must contain one point from Q. - Manfred Scheucher, Aug 30 2018
Also the number of internal edges of an (n+1)*(n+1) "square" of hexagons; i.e., n+1 rows, each of n+1 edge-adjacent hexagons, stacked with minimal overhang. - Jon Hart, Sep 29 2019
For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n+2; {1, 2n-1, 1, 1, 1, 2n-1, 1, 18n+4}]. - Magus K. Chu, Oct 13 2022

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
M. Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Leo Tavares, Illustration: Square Stars
Leo Tavares, Illustration: Split Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A001859. See Comments of A135713.
Cf. A000217, A000567, A001082, A002378, A016969, A049450, A174709.
Cf. second n-gonal numbers: A005449, A014105, A147875, A179986, A033954, A062728, A135705.
Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
Cf. A000290, A022144, A005563, A001105.
Cf. A056109.
Cf. A003154.

[n*(3*n+2) : n in [0..100]]; // Wesley Ivan Hurt, Sep 24 2017

Table[n*(3n+2), {n,0,60}] (* Harvey P. Dale, May 05 2011 *)
LinearRecurrence[{3,-3,1},{0,5,16},60] (* Harvey P. Dale, Jan 19 2016 *)
CoefficientList[Series[x*(5 + x)/(1 - x)^3,{x, 0, 60}], x] (* Stefano Spezia, Sep 01 2018 *)

a(n)=n*(3*n+2) \\ Charles R Greathouse IV, Nov 20 2012

O.g.f.: x*(5+x)/(1-x)^3. - R. J. Mathar, Jan 07 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=5, a(2)=16. - Harvey P. Dale, May 06 2011
a(n) = a(n-1) + 6*n - 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
For n > 0, a(n)^3 + (a(n)+1)^3 + ... + (a(n)+n)^3 + 2*A000217(n)^2 = (a(n) + n + 1)^3 + ... + (a(n) + 2n)^3; see also A033954. - Charlie Marion, Dec 08 2007
a(n) = Sum_{i=0..n-1} A016969(i) for n > 0. - Bruno Berselli, Jan 13 2011
a(n) = A174709(6*n+4). - Philippe Deléham, Mar 26 2013
a(n) = A001082(2*n). - Michael Turniansky, Aug 24 2013
Sum_{n>=1} 1/a(n) = (9 + sqrt(3)*Pi - 9*log(3))/12 = 0.3794906245574721941... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A002378(n) + A014105(n). - J. M. Bergot, Apr 24 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(12) - 3/4. - Amiram Eldar, Jul 03 2020
E.g.f.: exp(x)*x*(5 + 3*x). - Stefano Spezia, Jun 08 2021
From Leo Tavares, Oct 14 2021: (Start)
a(n) = A000290(n) + 4*A000217(n). See Square Stars illustration.
a(n) = A000567(n+2) - A022144(n+1)
a(n) = A005563(n) + A001105(n).
a(n) = A056109(n) - 1. (End)
From Leo Tavares, Oct 06 2022: (Start)
a(n) = A003154(n+1) - A000567(n+1). See Split Stars illustration.
a(n) = A014105(n) + 2*A000217(n). (End)

A049450 Pentagonal numbers multiplied by 2: a(n) = n(3n-1).

Original entry on oeis.org

0, 2, 10, 24, 44, 70, 102, 140, 184, 234, 290, 352, 420, 494, 574, 660, 752, 850, 954, 1064, 1180, 1302, 1430, 1564, 1704, 1850, 2002, 2160, 2324, 2494, 2670, 2852, 3040, 3234, 3434, 3640, 3852, 4070, 4294, 4524, 4760, 5002, 5250, 5504, 5764

Offset: 0

Joe Keane (jgk(AT)jgk.org)

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the sequence found by reading the line from 0 in the direction 0,2,.... The spiral begins:
.
                   56--55--54--53--52
                   /                 \
                 57  33--32--31--30  51
                 /   /             \   \
               58  34  16--15--14  29  50
               /   /   /         \   \   \
             59  35  17   5---4  13  28  49
             /   /   /   /     \   \   \   \
           60  36  18   6   0   3  12  27  48
           /   /   /   /   / . /   /   /   /
         61  37  19   7   1---2  11  26  47
           \   \   \   \       . /   /   /
           62  38  20   8---9--10  25  46
             \   \   \           . /   /
             63  39  21--22--23--24  45
               \   \               . /
               64  40--41--42--43--44
                 \                   .
                 65--66--67--68--69--70
(End)
Starting with offset 1 = binomial transform of [2, 8, 6, 0, 0, 0, ...]. - Gary W. Adamson, Jan 09 2009
Number of possible pawn moves on an (n+1) X (n+1) chessboard (n=>3). - Johannes W. Meijer, Feb 04 2010
a(n) = A069905(6n-1): Number of partitions of 6*n-1 into 3 parts. - Adi Dani, Jun 04 2011
Even octagonal numbers divided by 4. - Omar E. Pol, Aug 19 2011
Partial sums give A011379. - Omar E. Pol, Jan 12 2013
First differences are A016933; second differences equal 6. - Bob Selcoe, Apr 02 2015
For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n-2; {2, 2n-1, 6, 2n-1, 2, 18n-4}]. - Magus K. Chu, Oct 13 2022

			On a 4 X 4 chessboard pawns at the second row have (3+4+4+3) moves and pawns at the third row have (2+3+3+2) moves so a(3) = 24. - _Johannes W. Meijer_, Feb 04 2010
From _Adi Dani_, Jun 04 2011: (Start)
a(1)=2: the partitions of 6*1-1=5 into 3 parts are [1,1,3] and[1,2,2].
a(2)=10: the partitions of 6*2-1=11 into 3 parts are [1,1,9], [1,2,8], [1,3,7], [1,4,6], [1,5,5], [2,2,7], [2,3,6], [2,4,5], [3,3,5], and [3,4,4].
(End)
.
.                                                         o
.                                                       o o o
.                                      o              o o o o o
.                                    o o o          o o o o o o o
.                       o          o o o o o      o o o o o o o o o
.                     o o o      o o o o o o o    o o o o o o o o o
.            o      o o o o o    o o o o o o o    o o o o o o o o o
.          o o o    o o o o o    o o o o o o o    o o o o o o o o o
.    o     o o o    o o o o o    o o o o o o o    o o o o o o o o o
.    o     o o o    o o o o o    o o o o o o o    o o o o o o o o o
.    2      10         24             44                 70
- _Philippe Deléham_, Mar 30 2013

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014. (page 16)
Leo Tavares, Illustration: X Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567.
Bisection of A001859. Cf. A045944, A000326, A033579, A027599, A049451.
Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A035008 (Knight) and A002492 (Bishop).
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. [Bruno Berselli, Jun 10 2013]
Cf. sequences listed in A254963.
Cf. A003215, A016813, A000290, A000217.

List([0..50], n-> n*(3*n-1)); # G. C. Greubel, Aug 31 2019

[n*(3*n-1) : n in [0..50]]; // Wesley Ivan Hurt, Sep 24 2017

seq(n*(3*n-1),n=0..44); # Zerinvary Lajos, Jun 12 2007

Table[n(3n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,2,10},50] (* Harvey P. Dale, Jun 21 2014 *)
2*PolygonalNumber[5,Range[0,50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 01 2018 *)

a(n)=n*(3*n-1) \\ Charles R Greathouse IV, Nov 20 2012

[n*(3*n-1) for n in (0..50)] # G. C. Greubel, Aug 31 2019

O.g.f.: A(x) = 2*x*(1+2*x)/(1-x)^3.
a(n) = A049452(n) - A033428(n). - Zerinvary Lajos, Jun 12 2007
a(n) = 2*A000326(n), twice pentagonal numbers. - Omar E. Pol, May 14 2008
a(n) = A022264(n) - A000217(n). - Reinhard Zumkeller, Oct 09 2008
a(n) = a(n-1) + 6*n - 4 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010
a(n) = A014642(n)/4 = A033579(n)/2. - Omar E. Pol, Aug 19 2011
a(n) = A000290(n) + A000384(n) = A000217(n) + A000566(n). - Omar E. Pol, Jan 11 2013
a(n+1) = A014107(n+2) + A000290(n). - Philippe Deléham, Mar 30 2013
E.g.f.: x*(2 + 3*x)*exp(x). - Vincenzo Librandi, Apr 28 2016
a(n) = (2/3)*A000217(3*n-1). - Bruno Berselli, Feb 13 2017
a(n) = A002061(n) + A056220(n). - Bruce J. Nicholson, Sep 21 2017
From Amiram Eldar, Feb 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*log(3)/2 - Pi/(2*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(3) - 2*log(2). (End)
From Leo Tavares, Feb 23 2022: (Start)
a(n) = A003215(n) - A016813(n).
a(n) = 2*A000290(n) + 2*A000217(n-1). (End)

A006578 Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

Original entry on oeis.org

0, 1, 4, 8, 14, 21, 30, 40, 52, 65, 80, 96, 114, 133, 154, 176, 200, 225, 252, 280, 310, 341, 374, 408, 444, 481, 520, 560, 602, 645, 690, 736, 784, 833, 884, 936, 990, 1045, 1102, 1160, 1220, 1281, 1344, 1408, 1474, 1541, 1610, 1680, 1752, 1825, 1900, 1976, 2054

Offset: 0

N. J. A. Sloane

Equals (1, 2, 3, 4, ...) convolved with (1, 2, 1, 2, ...). a(4) = 14 = (1, 2, 3, 4) dot (2, 1, 2, 1) = (2 + 2 + 6 + 4). - Gary W. Adamson, May 01 2009
We observe that is the transform of A032766 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0, u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010
Equals row sums of a triangle with (1, 4, 7, 10, ...) in every column, shifted down twice for columns > 1. - Gary W. Adamson, Mar 03 2010
Number of pairs (x,y) with x in {0,...,n}, y odd in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012
Also A049451 and positives A000567 interleaved. - Omar E. Pol, Aug 03 2012
Similar to A001082. Members of this family are A093005, A210977, this sequence, A210978, A181995, A210981, A210982. - Omar E. Pol, Aug 09 2012

			G.f. = x + 4*x^2 + 8*x^3 + 14*x^4 + 21*x^5 + 30*x^6 + 40*x^7 + 52*x^8 + 65*x^9 + ...

Marc LeBrun, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Bruno Berselli, Hexagonal spiral containing the initial terms.
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche (2021) Vol. 76, No. 1, see p. 283.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A001859, A077043, A002620, A002378.
Row sums of A104567.
Cf. A000034, A032766, A002717, A070893. - Richard Choulet, Jan 28 2010
Cf. A051125.

[(6*n^2+4*n-1+(-1)^n)/8: n in [0..50] ]; // Vincenzo Librandi, Aug 20 2011

with (combinat): seq(count(Partition((3*n+1)), size=3), n=0..52); # Zerinvary Lajos, Mar 28 2008
# 2nd program
A006578 := proc(n)
    (6*n^2 + 4*n - 1 + (-1)^n)/8 ;
end proc: # R. J. Mathar, Apr 28 2017

Accumulate[LinearRecurrence[{1,1,-1}, {0,1,3}, 100]] (* Harvey P. Dale, Sep 29 2013 *)
a[ n_] := Quotient[n + 1, 2] (Quotient[n, 2] 3 + 1); (* Michael Somos, Jun 09 2014 *)
a[ n_] := Quotient[3 (n + 1)^2 + 1, 4] - (n + 1); (* Michael Somos, Jun 10 2015 *)
LinearRecurrence[{2, 0, -2, 1},{0, 1, 4, 8},53] (* Ray Chandler, Aug 03 2015 *)

{a(n) = (3*(n+1)^2 + 1)\4 - n - 1}; /* Michael Somos, Mar 10 2006 */

Expansion of x*(1+2*x) / ((1-x)^2*(1-x^2)). - Simon Plouffe in his 1992 dissertation
a(n) + A002620(n) = A002378(n) = n*(n+1).
Partial sums of A032766. - Paul Barry, May 30 2003
a(n) = a(n-1) + a(n-2) - a(n-3) + 3 = A002620(n) + A004526(n) = A001859(n) - A004526(n+1). - Henry Bottomley, Mar 08 2000
a(n) = (6*n^2 + 4*n - 1 + (-1)^n)/8. - Paul Barry, May 30 2003
a(n) = A001859(-1-n) for all n in Z. - Michael Somos, May 10 2006
a(n) = (A002378(n)/2 + A035608(n))/2. - Reinhard Zumkeller, Feb 07 2010
a(n) = (3*n^2 + 2*n - (n mod 2))/4. - Ctibor O. Zizka, Mar 11 2012
a(n) = Sum_{i=1..n} floor(3*i/2) = Sum_{i=0..n} (i + floor(i/2)). - Enrique Pérez Herrero, Apr 21 2012
a(n) = 3*n*(n+1)/2 - A001859(n). - Clark Kimberling, Jul 02 2012
a(n) = Sum_{i=1..n} (n - i + 1) * 2^( (i+1) mod 2 ). - Wesley Ivan Hurt, Mar 30 2014
a(n) = A002717(n) - A002717(n-1). - Michael Somos, Jun 09 2014
a(n) = Sum_{k=1..n} floor((n+k+1)/2). - Wesley Ivan Hurt, Mar 31 2017
a(n) = A002620(n+1)+2*A002620(n). - R. J. Mathar, Apr 28 2017
Sum_{n>=1} 1/a(n) = 3 - Pi/(4*sqrt(3)) - 3*log(3)/4. - Amiram Eldar, May 28 2022
E.g.f.: (x*(5 + 3*x)*cosh(x) - (1 - 5*x - 3*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Offset and description changed by N. J. A. Sloane, Nov 30 2006

A116940 Greatest m such that A116939(m) = n.

Original entry on oeis.org

0, 3, 6, 11, 16, 23, 30, 39, 48, 59, 70, 83, 96, 111, 126, 143, 160, 179, 198, 219, 240, 263, 286, 311, 336, 363, 390, 419, 448, 479, 510, 543, 576, 611, 646, 683, 720, 759, 798, 839, 880, 923, 966, 1011, 1056, 1103, 1150, 1199, 1248, 1299, 1350, 1403, 1456

Offset: 0

Reinhard Zumkeller, Feb 27 2006

From Andrew Rupinski, Nov 30 2009: (Start)
For n > 0, a(n) appears to be the set such that binomial(2*a(n),r) - binomial(2*a(n),r-2) = binomial(2*a(n),s) - binomial(2*a(n),s-2) for some r != s.
As a consequence of the Weyl Dimension Formula and the above comment, a(n) also appears to be the indices k such that the Symplectic Group Sp(k) has two fundamental irreducible representations of the same dimension. (End)

			a(n) = A000982(n) + A005843(n).
From _Andrew Rupinski_, Nov 30 2009: (Start)
a(1) = 3 and binomial(6,3)-binomial(6,1) = binomial(6,2)-binomial(6,0).
a(1) = 3 and the fundamental representations of Sp(3) are of dimensions 6, 14 and 14. a(2) = 6 and the fundamental representations of Sp(6) are of dimensions 12, 65, 208, 429, 572, and 429. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
John Tyler Rascoe, Illustration of the first 10 terms.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A116939.
Cf. A081352, A236283.
Cf. A028884, A099392.
Cf. A000217, A001859.

import Data.List (elemIndices)
a116940 n = last $ elemIndices n $ takeWhile (<= n + 1) a116939_list
-- Reinhard Zumkeller, Jun 28 2013

[(2*n*(n+4) -(-1)^n +1)/4: n in [0..55]]; // G. C. Greubel, Jan 26 2020

seq( (2*(n+2)^2 -(-1)^n -7)/4, n=0..55); # G. C. Greubel, Jan 26 2020

a = {0}; Do[AppendTo[a, If[Count[a, #-1] > #-1, #+1, #-1]] &@ a[[n]], {n, 1500}]; Most@ Values@ Map[Last, PositionIndex@ a] - 1 (* Michael De Vlieger, Dec 07 2016, Version 10 *)
Table[(2*(n+2)^2 -(-1)^n -7)/4, {n,0,55}] (* G. C. Greubel, Jan 26 2020 *)

vector(56, n, (2*(n+1)^2 +(-1)^n -7)/4) \\ G. C. Greubel, Jan 26 2020

[(2*n*(n+4) -(-1)^n +1)/4 for n in (0..55)] # G. C. Greubel, Jan 26 2020

a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 3.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Joerg Arndt, Apr 02 2011
G.f.: x*(3 - x^2)/((1 + x)*(1 - x)^3). - Arkadiusz Wesolowski, Jan 01 2012
a(n) = 2n + ceiling(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
a(n) = (2*n*(n + 4) - (-1)^n + 1)/4. - Bruno Berselli, Jun 14 2013
a(n) = A081352(n) - A236283(n + 1). - Miko Labalan, Dec 04 2016
From Klaus Purath, Jan 26 2020: (Start)
a(n) = binomial(n+2, 2) + floor((n-1)/2).
a(n) = floor(A028884(n)/2) - n.
a(n) = (n+1)^2 - A099392(n+1).
a(2*n)^2 - a(2*n-1)*a(2*n+1) = 3, n > 0.
a(2*n+1)^2 - a(2*n)*a(2*n+2) = (2*n+3)^2. (End)
E.g.f.: (1/2)*(x*(5 + x)*cosh(x) + (1 + 5*x + x^2)*sinh(x)). - Stefano Spezia, Jan 26 2020
a(n) = A000217(2*n) - 2*A001859(n-1) for n>0. - John Tyler Rascoe, Jul 31 2022
Sum_{n>=1} 1/a(n) = 11/8 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 16 2022

A359252 Number of vertices among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

Original entry on oeis.org

2, 13, 46, 101, 226, 417, 744, 1169, 1802, 2599, 3742, 5139, 7022, 9261, 12110, 15367, 19456, 24117, 29858, 36323, 43950, 52595, 62784, 73931, 86806, 101059, 117364, 135155, 155506

Offset: 2

Scott R. Shannon, Dec 22 2022

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

No formula for a(n) is currently known.

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 20.

Cf. A359253 (regions), A359254 (edges), A359258 (k-gons), A001859, A290447, A331702, A358746.

a(n) = A359254(n) - A359253(n) + 1 by Euler's formula.

A359253 Number of regions among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

Original entry on oeis.org

3, 14, 51, 116, 255, 466, 821, 1296, 2003, 2904, 4171, 5726, 7795, 10266, 13399, 17026, 21537, 26702, 32995, 40110, 48511, 57996, 69121, 81376, 95511, 111130, 128953, 148432, 170595

Offset: 2

Scott R. Shannon, Dec 22 2022

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

No formula for a(n) is currently known.

Scott R. Shannon, Image for n = 2. In this and other images the n points are shown as white dots.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 20.

Cf. A359252 (vertices), A359254 (edges), A359258 (k-gons), A001859, A290865, A359046, A358782.

a(n) = A359254(n) - A359252(n) + 1 by Euler's formula.

A359258 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

Original entry on oeis.org

3, 0, 8, 4, 2, 0, 22, 23, 4, 2, 0, 50, 52, 12, 2, 0, 110, 103, 36, 6, 0, 190, 200, 64, 12, 0, 314, 387, 88, 28, 4, 0, 498, 606, 152, 32, 8, 0, 770, 941, 228, 58, 4, 2, 0, 1132, 1352, 338, 68, 12, 2, 0, 1602, 1935, 532, 98, 4, 0, 2122, 2798, 684, 106, 16, 0, 2850, 3843, 940, 132, 24, 6

Offset: 2

Scott R. Shannon, Dec 23 2022

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

See A359252 and A359253 for other images of the circles.

			The table begins:
  3;
  0,     8,     4,    2;
  0,    22,    23,    4,   2;
  0,    50,    52,   12,   2;
  0,   110,   103,   36,   6;
  0,   190,   200,   64,  12;
  0,   314,   387,   88,  28,  4;
  0,   498,   606,  152,  32,  8;
  0,   770,   941,  228,  58,  4,  2;
  0,  1132,  1352,  338,  68, 12,  2;
  0,  1602,  1935,  532,  98,  4;
  0,  2122,  2798,  684, 106, 16;
  0,  2850,  3843,  940, 132, 24,  6;
  0,  3774,  4998, 1268, 192, 28,  6;
  0,  4950,  6475, 1644, 276, 44, 10;
  0,  6190,  8454, 1978, 326, 74,  4;
  0,  7778, 10737, 2520, 434, 52, 12, 4;
  0,  9674, 13224, 3202, 528, 58, 12, 4;
  0, 11978, 16169, 4116, 640, 68, 20, 4;
  ...

Scott R. Shannon, Image for n = 12.

Cf. A359253 (regions), A359252 (vertices), A359254 (edges), A001859, A332723, A359061, A359009.

Sum of row n = A359253(n);

A359254 Number of edges among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

Original entry on oeis.org

4, 26, 96, 216, 480, 882, 1564, 2464, 3804, 5502, 7912, 10864, 14816, 19526, 25508, 32392, 40992, 50818, 62852, 76432, 92460, 110590, 131904, 155306, 182316, 212188, 246316, 283586, 326100

Offset: 2

Scott R. Shannon, Dec 22 2022

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

See A359252 and A359253 for images of the circles.

Cf. A359252 (vertices), A359253 (regions), A359258 (k-gons), A001859, A290866, A359047, A358783.

a(n) = A359252(n) + A359253(n) - 1 by Euler's formula.

A359859 Number of vertices among all distinct circles that can be constructed from a 2 x n square grid of points using only a compass.

Original entry on oeis.org

2, 40, 190, 740, 1824, 3956, 7314, 12956, 20684, 32276, 47348, 68516, 94550, 128780, 170106, 222252, 283418, 358756, 445534, 550868, 670358, 811556, 970740, 1157168, 1363700, 1601384, 1864524, 2164668, 2493136, 2865176, 3269606, 3724112, 4215536, 4762284, 5353050

Offset: 1

Scott R. Shannon, Jan 16 2023

nonn

A circle is constructed for every pair of the 2 x n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the 2 x n points is conjectured to be 4*A001859(n-1).

No formula for a(n) is known.

Lucas A. Brown, Python program.
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 = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.

Cf. A359860 (regions), A359861 (edges), A359862 (k-gons), A001859, A359252.

a(n) = A359861(n) - A359860(n) + 1 by Euler's formula.

a(19)-a(35) from Lucas A. Brown, Oct 11 2024

cp's OEIS Frontend

A184010 n + floor(sqrt(-1+4n/3)); complement of A001859 (except for initial zero).

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A045944 Rhombic matchstick numbers: a(n) = n*(3*n+2).

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A049450 Pentagonal numbers multiplied by 2: a(n) = n*(3*n-1).

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A006578 Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

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A116940 Greatest m such that A116939(m) = n.

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A359252 Number of vertices among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

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A359253 Number of regions among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

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A045944 Rhombic matchstick numbers: a(n) = n(3n+2).

A049450 Pentagonal numbers multiplied by 2: a(n) = n(3n-1).