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A117625 Maximum number of regions defined by n zigzag-lines in the plane when a zigzag-line is defined as consisting of two parallel infinite half-lines joined by a straight line segment. 3
1, 2, 12, 31, 59, 96, 142, 197, 261, 334, 416, 507, 607, 716, 834, 961, 1097, 1242, 1396, 1559, 1731, 1912, 2102, 2301, 2509, 2726, 2952, 3187, 3431, 3684, 3946, 4217, 4497, 4786, 5084, 5391, 5707, 6032, 6366, 6709, 7061, 7422, 7792 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Note that the requirements imposed on the zigzag-line are neither the weakest nor the strongest imaginable. To relax the conditions, one might allow non-parallel half-lines. To strengthen them, one might demand the connecting line segment to be perpendicular to both half lines but still allow an arbitrary length of it, or go even further and additionally demand that all line segments be of equal length. The two latter cases would lend the problem a metrical nature.

REFERENCES

R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, 2nd Edition, p. 19, Addison-Wesley Publishing

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

Recurrence: a(n) = a(n-1) + 9*n - 8 Closed Form: a(n) = 4.5*n^2 - 3.5*n + 1

O.g.f: -(1-x+9*x^2)/(-1+x)^3 = -17/(-1+x)^2-9/(-1+x)^3-9/(-1+x) . - R. J. Mathar, Dec 05 2007

a(n) = (9*n^2-7*n+2)/2 = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jul 08 2012

EXAMPLE

a(0)= 1 because the plane is one region.

MAPLE

seq((9*k^2-7*k+2)/2, k=0..42);

MATHEMATICA

CoefficientList[Series[(1-x+9*x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 08 2012 *)

PROG

(MAGMA) [(9*n^2-7*n+2)/2: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

CROSSREFS

Cf. A000124.

Sequence in context: A116655 A085892 A101177 * A139323 A225525 A240395

Adjacent sequences:  A117622 A117623 A117624 * A117626 A117627 A117628

KEYWORD

easy,nonn

AUTHOR

Peter C. Heinig (algorithms(AT)gmx.de), Apr 08 2006

STATUS

approved

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Last modified October 1 22:19 EDT 2014. Contains 247522 sequences.