A239186 -id:A239186 - cp's OEIS frontend

A077591 Maximum number of regions into which the plane can be divided using n (concave) quadrilaterals.

1, 2, 18, 50, 98, 162, 242, 338, 450, 578, 722, 882, 1058, 1250, 1458, 1682, 1922, 2178, 2450, 2738, 3042, 3362, 3698, 4050, 4418, 4802, 5202, 5618, 6050, 6498, 6962, 7442, 7938, 8450, 8978, 9522, 10082, 10658, 11250, 11858, 12482, 13122, 13778

Offset: 0

Joshua Zucker and the Castilleja School MathCounts club, Nov 07 2002

Sequence found by reading the segment (1, 2) together with the line from 2, in the direction 2, 18, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 05 2011
For a(n) > 1, a(n) are the numbers such that phi(sum of the odd divisors of a(n)) = phi(sum of even divisors of a(n)). - Michel Lagneau, Sep 14 2011
Apart from first term, subsequence of A195605. - Bruno Berselli, Sep 21 2011
Engel expansion of 1F2(1; 1/2, 1/2; 1/8). - Benedict W. J. Irwin, Jun 21 2018
Let f(n) = 4*n^2 - 5, then (x, y, z) = (a(n+1), -f(n), -f(n + 1)) are solutions of the Diophantine equation x^3 + 4*y^3 + 4*z^3 = 512. - XU Pingya, Apr 25 2022

			a(2) = 18 if you draw two concave quadrilaterals such that all four sides of one cross all four sides of the other.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Keyang Li, when n=1, number of regions is 2; when n=2, maximum number of regions is 18
Keyang Li, when n=3, maximum number of regions is 50
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A006752, A077588, A239186.
Cf. A071974, A071975.
Cf. A016754, A069129.

Concatenation([1], List([1..2000], n->8*n^2 - 8*n + 2)); # Muniru A Asiru, Jan 29 2018

A077591:=n->`if`(n=0, 1, 8*n^2 - 8*n + 2); seq(A077591(n), n=0..50); # Wesley Ivan Hurt, Mar 12 2014

Table[2*(4*n^2 - 4*n + 1), {n,0,50}] (* G. C. Greubel, Jul 15 2017 *)

isok(n) = (sod = sumdiv(n, d, (d%2)*d)) && (sed = sumdiv(n, d, (1 - d%2)*d)) && (eulerphi(sod) == eulerphi(sed)); \\ from Michel Lagneau comment; Michel Marcus, Mar 15 2014

a(n) = 8*n^2 - 8*n + 2 = 2*(2*n-1)^2, n > 0, a(0)=1.
Proof from Keyang Li, Jun 18 2022: (Start)
Represent the configuration of n concave quadrilaterals by a planar graph with a node for each vertex of the quadrilaterals and for each intersection point. Let there be v_n nodes and e_n edges. By Euler's formula for planar graphs, a(n) = e_n - v_n + 2. When we go from n to n+1 quadrilaterals, each side of the new quadrilateral can meet each side of the existing quadrilaterals at most 4 times, so Dv_n := v_{n+1} - v_n <= 4*4n = 16n.
Each of these intersection points increases the number of edges in the graph by 2, so De_n := e_{n+1} - e_n = 4 + 2*Dv_n, Da_n := a(n+1) - a(n) = 4 + Dv_n <= 4+16*n.
These upper bounds can be achieved by taking n interwoven concave quadrilaterals (for n=1,2,3 see the attached Keyang Li links), and we achieve a(n) = 8n^2 - 8n + 2 (and v_n = 8n^2 - 4n, e_n = 4n*(4n-3)) for n > 0. QED (End)
For n > 0: A071974(a(n)) = 2*n+1, A071975(a(n)) = 2. - Reinhard Zumkeller, Jul 10 2011
a(n) = 1 + A069129(n), if n >= 1. - Omar E. Pol, Sep 05 2011
a(n) = 2*A016754(n-1), if n >= 1. - Omar E. Pol, Sep 05 2011
G.f.: (1 - x + 15*x^2 + x^3)/(1-x)^3. - Colin Barker, Feb 23 2012
E.g.f.: (8*x^2 + 2)*exp(x) - 1. - G. C. Greubel, Jul 15 2017
From Amiram Eldar, Jan 29 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = G/2, where G is Catalan constant (A006752).
Product_{n>=1} (1 + 1/a(n)) = cosh(Pi/sqrt(8)).
Product_{n>=1} (1 - 1/a(n)) = cos(Pi/sqrt(8)). (End)

A239667 Sum of the largest parts of the partitions of 4n into 4 parts.

Original entry on oeis.org

1, 17, 84, 262, 629, 1289, 2370, 4014, 6393, 9703, 14150, 19974, 27439, 36815, 48410, 62556, 79587, 99879, 123832, 151844, 184359, 221845, 264764, 313628, 368973, 431325, 501264, 579394, 666305, 762645, 869086, 986282, 1114949, 1255827, 1409634, 1577154, 1759195, 1956539, 2170038, 2400568

Offset: 1

Wesley Ivan Hurt and Antonio Osorio, Mar 23 2014

			Add the numbers in the first column for a(n):
                                             13 + 1 + 1 + 1
                                             12 + 2 + 1 + 1
                                             11 + 3 + 1 + 1
                                             10 + 4 + 1 + 1
                                              9 + 5 + 1 + 1
                                              8 + 6 + 1 + 1
                                              7 + 7 + 1 + 1
                                             11 + 2 + 2 + 1
                                             10 + 3 + 2 + 1
                                              9 + 4 + 2 + 1
                                              8 + 5 + 2 + 1
                                              7 + 6 + 2 + 1
                                              9 + 3 + 3 + 1
                                              8 + 4 + 3 + 1
                                              7 + 5 + 3 + 1
                                              6 + 6 + 3 + 1
                                              7 + 4 + 4 + 1
                                              6 + 5 + 4 + 1
                                              5 + 5 + 5 + 1
                              9 + 1 + 1 + 1  10 + 2 + 2 + 2
                              8 + 2 + 1 + 1   9 + 3 + 2 + 2
                              7 + 3 + 1 + 1   8 + 4 + 2 + 2
                              6 + 4 + 1 + 1   7 + 5 + 2 + 2
                              5 + 5 + 1 + 1   6 + 6 + 2 + 2
                              7 + 2 + 2 + 1   8 + 3 + 3 + 2
                              6 + 3 + 2 + 1   7 + 4 + 3 + 2
                              5 + 4 + 2 + 1   6 + 5 + 3 + 2
                              5 + 3 + 3 + 1   6 + 4 + 4 + 2
                              4 + 4 + 3 + 1   5 + 5 + 4 + 2
               5 + 1 + 1 + 1  6 + 2 + 2 + 2   7 + 3 + 3 + 3
               4 + 2 + 1 + 1  5 + 3 + 2 + 2   6 + 4 + 3 + 3
               3 + 3 + 1 + 1  4 + 4 + 2 + 2   5 + 5 + 3 + 3
               3 + 2 + 2 + 1  4 + 3 + 3 + 2   5 + 4 + 4 + 3
1 + 1 + 1 + 1  2 + 2 + 2 + 2  3 + 3 + 3 + 3   4 + 4 + 4 + 4
    4(1)            4(2)           4(3)            4(4)       ..   4n
------------------------------------------------------------------------
     1               17             84             262        ..   a(n)

A. Osorio, A Sequential Allocation Problem: The Asymptotic Distribution of Resources, Munich Personal RePEc Archive, 2014.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-6,6,-3,3,-3,1).

Cf. A238328, A238340, A238702, A238705, A238706, A239056, A239057, A239059, A239186.

I:=[1,17,84,262,629,1289,2370,4014,6393]; [n le 9 select I[n] else 3*Self(n-1)-3*Self(n-2)+3*Self(n-3)-6*Self(n-4)+6*Self(n-5)-3*Self(n-6)+3*Self(n-7)-3*Self(n-8)+Self(n-9): n in [1..45]]; // Vincenzo Librandi, Aug 29 2015

CoefficientList[Series[-(9*x^6 + 32*x^5 + 50*x^4 + 58*x^3 + 36*x^2 + 14*x +
1)/((x - 1)^5*(x^2 + x + 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 13 2014 *)
LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {1, 17, 84, 262, 629, 1289, 2370, 4014, 6393}, 50](* Vincenzo Librandi, Aug 29 2015 *)
Table[Total[IntegerPartitions[4 n,{4}][[All,1]]],{n,40}] (* Harvey P. Dale, Apr 25 2020 *)

Vec(-x*(9*x^6+32*x^5+50*x^4+58*x^3+36*x^2+14*x+1) / ((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Mar 23 2014

G.f.: -x*(9*x^6+32*x^5+50*x^4+58*x^3+36*x^2+14*x+1) / ((x-1)^5*(x^2+x+1)^2). - Colin Barker, Mar 23 2014

Let b(1) = 4, with b(n) = (n/(n-1)) * b(n-1) + 4n * Sum_{i=0..2n} (floor((4n-2-i)/2)-i) * floor((sign(floor((4n-2-i)/2)-i)+2)/2). Then a(1) = 1, with a(n) = a(n-1) + b(n-1)/(4n-4) + Sum_{i=j+1..floor((4n-2-j)/2)} ( Sum_{j=0..2n} (4n-2-i-j) * floor((sign(floor((4n-2-j)/2)-j)+2)/2) ). - Wesley Ivan Hurt, Jun 13 2014

A239195 Sum of the next to smallest parts in the partitions of 4n into 4 parts with smallest part = 1.

Original entry on oeis.org

1, 5, 17, 42, 78, 134, 215, 315, 447, 616, 812, 1052, 1341, 1665, 2045, 2486, 2970, 3522, 4147, 4823, 5579, 6420, 7320, 8312, 9401, 10557, 11817, 13186, 14630, 16190, 17871, 19635, 21527, 23552, 25668, 27924, 30325, 32825, 35477, 38286, 41202, 44282, 47531

Offset: 1

Wesley Ivan Hurt, Mar 11 2014

			For a(n) add the numbers in the third columns.
                                               13+ 1 + 1 + 1
                                               12+ 2 + 1 + 1
                                               11+ 3 + 1 + 1
                                               10+ 4 + 1 + 1
                                               9 + 5 + 1 + 1
                                               8 + 6 + 1 + 1
                                               7 + 7 + 1 + 1
                                               11+ 2 + 2 + 1
                                               10+ 3 + 2 + 1
                              9 + 1 + 1 + 1    9 + 4 + 2 + 1
                              8 + 2 + 1 + 1    8 + 5 + 2 + 1
                              7 + 3 + 1 + 1    7 + 6 + 2 + 1
                              6 + 4 + 1 + 1    9 + 3 + 3 + 1
                              5 + 5 + 1 + 1    8 + 4 + 3 + 1
                              7 + 2 + 2 + 1    7 + 5 + 3 + 1
               5 + 1 + 1 + 1  6 + 3 + 2 + 1    6 + 6 + 3 + 1
               4 + 2 + 1 + 1  5 + 4 + 2 + 1    7 + 4 + 4 + 1
               3 + 3 + 1 + 1  5 + 3 + 3 + 1    6 + 5 + 4 + 1
1 + 1 + 1 + 1  3 + 2 + 2 + 1  4 + 4 + 3 + 1    5 + 5 + 5 + 1
    4(1)            4(2)           4(3)            4(4)       ..   4n
------------------------------------------------------------------------
     1               5              17              42        ..   a(n)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A238328, A238340, A238702, A238705, A238706, A239056, A239057, A239059, A239186.

b[n_] := Sum[(((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i)) - ((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i)) - ((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i))/(4 n)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; Table[b[n], {n, 50}]
LinearRecurrence[{2,-1,2,-4,2,-1,2,-1},{1,5,17,42,78,134,215,315},60] (* Harvey P. Dale, Jul 05 2025 *)

Vec(x*(4*x^5+5*x^4+11*x^3+8*x^2+3*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014

G.f.: x*(4*x^5+5*x^4+11*x^3+8*x^2+3*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 12 2014

a(n) = 2*a(n-1)-a(n-2)+2*a(n-3)-4*a(n-4)+2*a(n-5)-a(n-6)+2*a(n-7)-a(n-8). - Wesley Ivan Hurt, Jul 08 2025

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A077591 Maximum number of regions into which the plane can be divided using n (concave) quadrilaterals.

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A239667 Sum of the largest parts of the partitions of 4n into 4 parts.

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A239195 Sum of the next to smallest parts in the partitions of 4n into 4 parts with smallest part = 1.

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