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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. 5
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
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
(PARI) a(n)=n*(9*n-7)/2+1 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
Cf. A000124.
Sequence in context: A361760 A085892 A101177 * A297763 A254962 A139323
KEYWORD
easy,nonn
AUTHOR
Peter C. Heinig (algorithms(AT)gmx.de), Apr 08 2006
STATUS
approved

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Last modified May 28 11:56 EDT 2024. Contains 372913 sequences. (Running on oeis4.)