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A077591 Maximum number of regions into which the plane can be divided using n (concave) quadrilaterals. 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For n > 0: A071974(a(n)) = 2*n+1, A071975(a(n)) = 2. - Reinhard Zumkeller, Jul 10 2011

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

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

Engel expansion of 1F2(1;1/2,1/2;1/8). - Benedict W. J. Irwin, Jun 21 2018

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

FORMULA

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]

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

(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

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

CROSSREFS

Cf. A077588, A239186.

Sequence in context: A208652 A223469 A048910 * A050808 A058653 A058794

Adjacent sequences:  A077588 A077589 A077590 * A077592 A077593 A077594

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified July 21 03:29 EDT 2018. Contains 312842 sequences. (Running on oeis4.)