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A077591 Maximum number of regions into which the plane can be divided using n (concave) quadrilaterals. 12

%I

%S 1,2,18,50,98,162,242,338,450,578,722,882,1058,1250,1458,1682,1922,

%T 2178,2450,2738,3042,3362,3698,4050,4418,4802,5202,5618,6050,6498,

%U 6962,7442,7938,8450,8978,9522,10082,10658,11250,11858,12482,13122,13778

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

%C For n > 0: A071974(a(n)) = 2*n+1, A071975(a(n)) = 2. - _Reinhard Zumkeller_, Jul 10 2011

%C 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

%C 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

%C Apart from first term, subsequence of A195605. - _Bruno Berselli_, Sep 21 2011

%C For n>3, a(n) is the fourth least number k = x + y, with x>0 and y>0, such that there are n different pairs (x,y) for which x*y/k is an integer. - _Paolo P. Lava_, Jan 29 2018

%C Engel expansion of 1F2(1;1/2,1/2;1/8). - _Benedict W. J. Irwin_, Jun 21 2018

%H Vincenzo Librandi, <a href="/A077591/b077591.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = 8*n^2 - 8*n + 2 = 2*(2*n-1)^2, n>0, a(0)=1. [It would be nice to have a proof, or even a reference to a proof. - _N. J. A. Sloane_, Nov 30 2017]

%F a(n) = 1 + A069129(n), if n >= 1. - _Omar E. Pol_, Sep 05 2011

%F a(n) = 2*A016754(n-1), if n >= 1. - _Omar E. Pol_, Sep 05 2011

%F G.f.: (1-x+15*x^2+x^3)/(1-x)^3. - _Colin Barker_, Feb 23 2012

%F E.g.f.: (8*x^2 + 2)*exp(x) -1. - _G. C. Greubel_, Jul 15 2017

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

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

%t Table[2*(4*n^2 - 4*n + 1), {n,0,50}] (* _G. C. Greubel_, Jul 15 2017 *)

%o (PARI) 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

%o (GAP) Concatenation([1], List([1..2000], n->8*n^2 - 8*n + 2)); # _Muniru A Asiru_, Jan 29 2018

%Y Cf. A077588, A239186.

%K nonn,easy

%O 0,2

%A _Joshua Zucker_ and the Castilleja School MathCounts club, Nov 07 2002

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Last modified April 18 15:05 EDT 2019. Contains 322209 sequences. (Running on oeis4.)