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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 Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Index entries for 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 (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 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 December 9 15:36 EST 2023. Contains 367693 sequences. (Running on oeis4.)